LIBRARY
OF THE
University of California.
Class
.y^i
^'"t-k
'-■J 'r ■,-».-• . .
Ff^^J^:-:. ....
'^•-*:.:
■m^
" ■:,s>i.
'■-<^^.
vJ».v
r-.''.*-'^''f-..
AN INTRODUCTION
TO THE
INFINITESIMAL CALCULUS
NOTES FOR THE USE OF SCIENCE
AND ENGINEERING STUDENTS
BY
H. S. CARSLAW, M.A., D.Sc, F.R.S.E.
n
PROFESSOR OF MATHEMATtCS IN THE UNIVERSITY OF SYDNEY
FORMERLY FELLOW OF EMMANUEL COLLEGE, CAMBRIDGE
AND LECTURER IN MATHEMATICS AT THE UNIVERSITY OF GLASGOW
OF THE
UNIVERSITY
OF
LONGMANS, GREEN, AND CO.
39 PATERNOSTER ROW, LONDON
NEW YORK, AND BOMBAY
1905
Qf\3o3
03?
t-'a'f/jo
PREFACE
These introductory chapters in the Infinitesimal Calculus were
lithographed and issued to the students of the First Year in
Science and Engineering of the University of Sydney at the
beginning of last session. They form an outline of, and were
meant to be used in conjunction with, the course on The Elements
of Analytical Geometry and the Infinitesimal Calculus, which leads
up to a term's work on Elementary Dynamics.
The standard text-books amply suffice for the detailed study
of this subject in the second year, but the absence of any dis-
cussion of the elements and first principles suitable foi- the first
year work, was found to be a serious hindrance to the work of
the class. For such students a separate course on Analytical
Geometry, Avithout the aid of the Calculus, is not necessary, and
the exclusion of the methods of the Calculus from the analytical
study of the Conic Sections is quite opposed to the present
unanimous opinion on the education of the engineer. It has
been our object to present the fundamental ideas of the Calculus
in a simple manner and to illustrate them by practical examples,
and thus to enable these students to use its methods intelli-
gently and readily in their Geometrical, Dynamical, and Physical
work early in their University course. This little book is not
meant to take the place of the standard treatises on the subject,
and, for that reason, no attempt is made to do more than give
the lines of the proof of some of the later theorems. As an
introduction to these works, and as a special text-book for such
208534
vi PREFACE
a " short course " as is found necessary in the engineering schools
of the Universities and in the Technical Colleges, it is hoped that
it may be of some value.
In the preparation of these pages I have examined most of
the standard treatises on the subject. To Nernst and Schonflies'
Lehrbuch der Differential- und Integral - Reclmung, to Vivanti's
Complementi di Matematica ad uso dei Chemici e del Naturalisti, to
Lamb's Infinitesimal Calculus, and to Gibson's Elementary Treatise
on the Calculus, I am conscious of deep obligations. I should
also add that from the two last-named books, and from those
of Lodge, Mellor, and Murray, many of the examples have been
obtained.
In conclusion, I desire to tender mj' thanks to my Colleagues
in the University of Sydney, Mr. A. Xewham and Mr. E. M.
Moors, for assistance in reading the proof-sheets ; to my students,
Mr. D. R. Barry and Mr. R. J. Lyons, for the verification of
the examples ; also to my old teacher, Professor Jack of the
University of Glasgow, and to Mr. D. K. Picken and Mr. R. J.
T. Bell of the Mathematical Department of that University, by
whom the final proofs have been revised.
H. S. CARSLAW.
The University of Sydney,
Jime 1905.
CONTENTS
CHAPTEE T
CO-ORDINATE GEOMETRY -THE STRAIGHT LIXE
SECT. PAGE
1. Cartesian Co-ordinates . . . . .1
2. The Co-ordinates of the Point at which a Line is divided in
a "iven Ratio . . . . . .1
3. The Equation of the First Degree . . . .3
4. Lines whose Equations are given . . . .4
5. The Gradient of a Straight Line . . . .5
6. Different Forms of the Equation of the Straight Line . 5
7. The Perpendicular Form . . . . .7
8. The Point of Intersection of two Straight Lines . . 7
9. The Angle between two Straight Lines . . .8
10. The Length of the Perpendicular from a given Point upon a
Straight Line . . . . . .10
Examples on Chapter I . . . .12
CHAPTEE II
THE MEANING OF DIFFERENTIATION
11. The Idea of a Function . . . . .14
12. Illustrations from Physics and Dynamics . . .14
13. The Fundamental Problem of the Differential Calculus . 16
vii
via
CONTENTS
SECT.
14. Rectilinear Motion
PAGE
16
15. Limits. Dift'erential Coefficient .17
16. Geometrical Illustration of the Meaning of the Differential
Coefficient . . . . . .20
17. Apjjroximate Graphical Determination of the Differential
Coefficient . . . . . .21
18. The Shape of the Curve y =/{.':) deduced from the Differential
Coefficient of/(a;) . . . . .22
Examples on Chapter II.
23
CHAPTER III
DIFFEREXTIATIOX OF ALGEBRAIC FUNCTIONS; AND SOME
GENERAL THEOREMS IN DIFFERENTIATION
19. Differentiation of cc" . . . . .25
20. Some General Theorems —
I. Differentiation of a Constant . .26
II. Differentiation of the Product of a Con.stant and a
Function . . . .26
III. Differentiation of a Sum . . .27
IV. Differentiation of the Product of Two Functions . 27
V. Differentiation of the Quotient of Two Functions 28
VI. Differentiation of a Function of a Function . 29
Examples on Chapter III. . . .31
CHAPTER IV
THE DIFFERENTIATION OF THE TRIGONOMETRIC FUNCTIONS
21. Differentiation of sin ;/■
22. Differentiation of cos .'■ . . ,
23. Differentiation of tan ,'■
24. Geometrical Proofs of these Theorem.-
25. Difl'erentiatinn of pin"Vv .
33
34
34
35
36
CONTENTS ix
SECT. PAGE
26. Differentiation of COS" ^x . . .36
27. Differentiation of tan "^x . . .37
Examples ox Chaptek IV. . .38
CHAPTEE V
THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS; MAXIMA
AND MINIMA ; PARTIAL DIFFERENTIATION
28. Introductory ....
29. Dift'erentiation of e^
30. Differentiation of log x .
31. Logarithmic Differentiation
32. Differentiation of e""'' sin 6.r
33. Maxima and Minima of Functions of one Variable
34. Points of Inflection
35. Partial Differentiation
36. Partial Differentiation {continued) .
37. Total Differentiation
38. Differentials ....
Examples on Chapter V.
40
40
41
42
42
43
45
45
46
47
48
50
CHAPTEE VI
THE CONIC SECTIONS
39. Introductory . . . . .53
40. Discussion of the Parabola and Examples . . .53
41. Discussion of the Ellipse and Examples . .56
42. Discussion of the Hyperbola and Examples . . 59
CHAPTEE VII
THE INTEGRAL CALCULUS— INTEGRATION
43. Definition of the Indefinite Integral . - .64
44. Standard Integrals . . . . .65
CONTENTS
SECT.
45. Two General Theorems
46. Integration by Substitution
47. Integration by Substitution {continued)
48. Integration by Part.s
Examples on Chapter VII.
66
67
69
72
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
CHAPTEE YIII
THE DEFIXITE INTEGRAL AXD ITS APPLICATIOXS
Introductory ......
Areas of Curves. The Definite Integral as an Expression
for the Area ......
The Definite Integral as the Limit of a Sum
The Evaluation of the Definite Integral from its Definition
as the Limit of a Sum .....
Properties of
J{x)dx
XO
Application of the Definite Integral to Areas in Polar
Co-ordinates ......
Application of the Definite Integral to Lengths of Curves
Application of the Definite Integral to Volumes of Solids
Application of the Definite Integral to Surfaces of Solids of
Revolution ......
Application of the Definite Integral to the Centre of Gra\-ity
of a Solid Body .....
Application of the Definite Integral to the Moment of
Inertia of a Solid Body ....
Examples on Chapter VIII. ....
76
79
81
83
84
85
86
87
88
90
91
Answers
95
Of THE
UNIVERSITY
OF
CHAPTEE I
THE ANALYTICAL GEOMETRY OF THE STRAIGHT LINE*
§ 1. Cartesian Co-ordinates.
The position of a point on a plane may be fixed in ditt'erent
ways. In particular it is determined if its distances from two
fixed perpendicular lines in the plane are known, the usual con-
ventions with regard to sign lieing adopted. These two line.^
Ox and 0^ are called the axes of x and y ; and the lengths OM
and OX, which the perpendiculars from the point P cut off from
the axes, are called the co-ordinates of the point P and denoted
by X and //. OM and OX are taken positive or negative accord-
ing as they are measured along Ox and Oij, or in the opposite
directions.
Ex. 1. ilark on a piece of squared paper the position of the points
(±2, ±3).
2. Prove that the distance between the points (2, -3), and (-2, -3) is
■2\/l3.
3. Prove that tlie distance d between the points {xi, t/i), (.v.i, 2/2) i'' given
by d- = {Xi- x^f 4- (yj - y.,f.
4. Prove that the co-ordinates of any point (.r, y) upon the circle whose
centre is at the point (a, h) and whose radius is r, satisfy the equation
^ 2. The Co-ordinates of a Point dividing the Line joining two
given Points in a given Ratio 1 : m.
Let P^ and 1'., be the two given points {j\, y^, (./■.„ //.,) ; and
let P (.r, y) divide P^P., in the ratio / : m (see Fig. 1).
* The student is recommended to read pp. 1-25 of Hall's Introduction to
Graphical Algebra (2nd ed.) before commencing this work.
0
THE ANALYTICAL GEOMETRY
Draw P^Mp PM, and P^Mo perpendicular to Ox ; P^HK and
PL parallel to O.i', meeting PJNI and P^M^ in H, K, and L.
Since
Similarly
M,
y
Fig. 1.
P,H _ P^P
PL ~PP,
m
I
tK' tC-i
x^- X m
x(l -t- m) = lx.2 + mx^,
lx„ + nix.
x =
I + m
I + m
M Mj
These are the co-ordinates of the internal point of section.
Those of the external point may he found in the same way to be
IXn
X=^
■mx-,
l-m
OF THE STRAIGHT LINE
and
y =
^2 - m/i
I - m
Ex. 1. Prove that the co-ordinates of the middle point of the line whicii
cuts off unit length from Oa; and 0// are h and |.
2. Find the co-ordinates of the jioints of trisection of this line, and also of
the points which divide it externally in the ratio 1 : 2.
3. Prove that the C.G. of the triangle whose angular ]ioints are (2, 1),
(4, 3), (2, 5) is the point (-, 3); and give the general the
leorem.
§ 3. The Equation of the First Degree represents a Straight
Line.
If the point P move along a curve the co-ordinates of the
point are not independent of each other. In mathematical
language " y is a function of z," and we speak of y =/(■''■) as the
equation of the curve, meaning that this equation is satisfied by
the co-ordinates (x, y) of any point upon the curve. For example,
the equation of the circle whose centre is at the origin and
whose radius is a is *- + y^ = a-. The properties of curves may
often be obtained by discussing their equations.
The simplest equation is that of the first degree, ax + hy + r -0,
It, b, and c being constants.
For example, take the equation
X +2y=4:.
By assigning any value to x and solving
the equation for y we obtain, as in the
accompanying table, the co-ordinates of any
number of points upon the locus. Plotting
these points on the diagram we see that
all lie upon a straight line.
AVe proceed to prove that this is true
in general ; in other words, that all the
points whose co-ordinates satisfy the equation
ax + by + c = 0
lie upon a straight line.
Let P^ (Xp y^ and P.^ {x.^, y.,) be two points upon the locus.
Then we have ax^ + by^ + c = 0 . . . ( 1 )
aXo + by^ + c^O ... (2)
X
V
-3
3-5
_ 2
3
- 1
2 -.5
0
0
1
1-5
2
1
3
•5
4 THE ANALYTICAL GEOMETRY
Multiplying (1) by in and (2) by I, and adding, we obtain
a{b:, + m,i\) + h{ly.^ + mii^) + (■(/ + m) = 0,
\ I + 7n / \ I + m /
-^ ^ Ix^'rmx^ ly„ + my, . .. . ,
i5ut — ^ -, ~ — are the co-ordinates of the point
/ + III I + m
dividing P-^P^ in the ratio I : in, and /, /// may be chosen at
random. It follows that if Pp P.^ are two fixed points on the
locus given by
a:c + by + c == 0,
any other point on the unlimited straight line P^P., is also upon
the locus ; and it can easily be shown that no point off this line
lies upon the curve.
Therefore the equation
a.r + by + c = 0
represents a straight line.
Ex. Prove this theorem by showing that if PQR are any three points
whose co-ordinates satisfy the given equation, the triangle PQR has zero area.
^ 4. In the last article we have shown that the equation of
the first degree represents a straight line. It is not then
necessary in plotting the locus given by such an equation to
proceed as we did above in the example x + 2y - 4. Two points
fix a straight line. Therefore we have only to find two points
whose co-ordinates satisfy the equation. The most convenient
points are those where the line cuts the axes, and these are
found by putting x = 0 and y = 0, respectively, in the equation.
Ex. 1. Draw tlie lines (i.) ^' = 0, x = l, ;r= - 1
(ii.) 7/ = 0, y = 2, y=-2
(iii.) r + y=0, x + y^l
(iv.) y = 2x, //=2,r-r3
2. Determine whether the point (2, 3) is on tlie line
4x + 3y=l5.
3. What is the condition that the point («, h) should lie upon the line
ax + by = 2ab '{
OF THE STRAIGHT LINE
§ 5. The Gradient of a Line.
When we speak of "the gradient" of a road being 1 in 200
we usually mean that the ascent is 1 foot vertical for 200 feet
horizontal. This might also be called the slope of the road.
The same expression is used with regard to the straight line.
The "gradient" or the "slope" of a straight line is its rise per
unit horizontal distance ; or the ratio of the increase in y to the
increase in x as we move along the line. This is evidently the
same at all points of the
straight line, and is equal
to the tangent of the angle
the line makes with the
axis of ;'; measured in the
positive direction.
To save ambiguity it
is well to fix upon the
angle to be chosen, and
in these pages it will be
convenient to consider
the line as always drawn
upward in the direction
of the arrow (Fig. 2), and thus to restrict the angle (/> to lie
between 0° and 180^
Wiien 0<(^<^ the gradient is positive.
When ^ <(/)<- the gradient is negative.
Ex. 1. Write down the values of <p for the lines in § 4 (i.).
2. Prove that tlie gradient of the line y = m.v + c is m, and interpret the
constant c.
§ 6. Different forms of the Equation of the Straight Line.
In the preceding articles we have shown that the equation
ax + 1)1/ + c = 0
represents a straight line, and we have seen how the line may
be drawn when its equation is given. We have now to show
how to obfain the equation of the line when its position is given.
(A) The equation of the line fhrouf/h two given points.
Fig. 2.
6 THE ANALYTICAL GEOMETEY
Let (a-p //^), (.7;.,, y.^ be the two given points. Let (.';, y) be
the co-ordinates of any jjoint upon the line. Then it is clear
(cf. Fig. 1) that
•^ — ^1 = the gradient of the line,
and that Vilih =
.Xg — X-^
Thus we have the equation
yjlllJ-hZll
«// ^~ tC-i •^o t'j-*
between the co-ordinates (a-, y) of the representative point and
the co-ordinates (.r^ y^ (.r.,, y,^ of the fixed points. This is the
equation of the straight line through these points. It is more
conveniently written
X - x^_ y - y^
^1 - ^2 ~ Vi- y-2
It folloAvs that
(B) The equation of the line through [u-^, y^), making an angle ^
with the axis of x, is
qc /yi ' ^
and that
(C) The equation of the line which cuts off a length c from the axis
of y, and is inclined at an angle whose tangent is in to the axis ofx, is
y = mx + c,
and that
(D) The equation of the line which cuts off' intercepts a and h from
the axis of x and y is
%f=l.
a b
Ex. 1. Write down the equations of the lines through the following pairs
of points : (1, 1) (1, - 1) ; (1, 2) ( - 1, - 2) ; (3, 4) {5, 6) ; {a, b) {a, - h). '
2. Find the equations of the lines tlirough the point (3, 4) with gradient
±5, and draw the lines.
3. The lines y = x and y = 2x form two adjacent sides of a parallelogram,
the <)2)posite angular point liciiig (4, 5). Find the equations of the other two
sides ; and of the diagonals.
OF THE STRAIGHT LINE 7
4. Write down the equations of the lines making angles 30°, 45°, 60°,
120°, 135°, and 150° with the axis of x, which cut this axis at unit distance
from the origin in the negative direction.
§7. The "Perpendicular" Form of the Equation of the
Straight Line.
A straight line is determined when the length of the perpen-
dicular upon it from the
origin, and the direction
of this perpendicular are
given.
Let ON be the perpen-
dicular, ]), upon the line.
Let the angle between
ON and Ox be a, this
angle lying between 0
and 27r (cf. Fig. 3).
Then N is the point
, • X Fig. 3.
(|> cos a, p Sin a).
Using the form (B) of § 6 the equation of the line becomes
y - V sm a
- — = tan d) = tan a + „
x-p cos a \ 2
cos a
sin a
This reduces to
(E) X cos a + y sin a =7/.
N.B. — The quantity p is to be taken always positive, and the
angle a is the angle between 0.^ and ON.
§ 8. The Point of Intersection of Two Straight Lines.
Since the point of intersection of the two lines
ax + by + (■ - 0
ax + b'y + c - 0
lies on both lines, its co-ordinates x, y satisfy both equations.
Solving the equations we have
X
V
1
ic' - h'r m — c'a ah' — a'l
8 THE ANALYTICAL GEOMETRY
It is clear that if
ah' - ah = 0,
and neither of the other two denominators vanish, the co-ordinates
.T, y are infinite, and the lines are parallel.
If in addition
ca' - c'a = 0
, a h c
we have - = - = -
a b c
and the third denominator he - h'c also vanishes.
In this case the two equations are not independent, and they
really represent the same straight line.
Ex. 1. Find the co-ordinates of the point of intersection of the line.s
.c + 2y=&.
Illustrate your result by a diagram.
2. Find the equations of the lines through (2, 3) parallel to
3. Find the co-ordinates of the angular points of the triangle whose sides
are given by
^+ y = 2 . . . . . (1),
Sx-2y=l (2),
ix+Bi/=2-l (3).
Also find the equations of the medians of this triangle and the co-ordinates
of its C.G.
§ 9. The Angle between Two Straight Lines whose Equations
are given.
The equations of the lines may always be reduced to the forms
( 1 ) If = inx + c,
(2) y=m'.r + c',
and in this case the angles they make with the axis of x are f/j
and (^' where
tan (^ = m, (cf. Fig. 4)
tan </)' = m'.
Hence
tan ((^ - (/, = r = tan d,
1 -f- mm
and the angle 6 between the lines is tan ( ,
OF THE STRAIGHT LINE
9
Unless care is shown in taking for the line (1) that with the
greatei^ slope, Ave would obtain a negative value for the tangent
of the an trie between the lines. The reason for this is obvious.
Fig. 4.
It follows that
(i.) The lines are parallel if m = m ;
(ii.) The lines are perpendicular if mm +1 = 0.
When the equations are
ax + hij + c — 0
a'x + h'!J + r,' = 0
(i.) The lines are parallel if — = p .; -y
(ii.) The lines are perpendicular if aa + hh' = 0.
Ex. 1. Write down the equation of the straight liue through (1, 2)
perpendicular to x-y = 0.
2. Find the angles between the lines
..•-2y+l=0}
x + By + 2 = 0)
and 4a; + 3j/=12)
3,'+ 4?/ =12/'
and draw the lines.
10
THE ANALYTICAL GP:OMETRY
3. Write down the equation of the straight line through (a, b) perpen-
dicular to bx-ay= a^ + IP'.
4. Write down the equation of the line bisecting the line joining (1, 2)
(3, 4) at right angles, and the equations of the perpendiculars upon both lines
from the origin.
5. Prove that l{x-a)-^m{y-h) = () is a line through (a, &) parallel to
lc + viij = 0 ; and write down the equation of the line through (a, h) perpen-
dicular to lx + my = 0.
6. AVrite down the equations of the lines through the C.G. of the triangle
whose angular points are at (4, -5) (5, -6) (3, 1) parallel and perpendicular
to the sides.
§ 10. The Length of the Perpendicular from a Point (.i^, v/^,)
upon a Straight Line whose Equation is given.
(i.) If the equation of
the straight line is given
in the " perpendicular "
form
X cos a + y sill u =J) (1),
the line through P (.Tq, y^)
parallel to it is given
Fio.5. (^^ (2) by
{x - .Tq) cos a-T (y - y^ sin a = 0
or X cos a + y sin a = Xq cos a + yQ sin a.
(2),
But if Pq is the perpendicular ONq from 0 upon the line (2),
and if N, Nq are on the same side of 0, the equation of PN^ may
be written
X cos a + // sin a =2\r
Therefore
X(^ cos a + v/p sin a =2\y
Also the perpendicular from P upon the line (1) is
ONp - ON, (cf. Fig. 5)
ie. i>^-p,
i.e. Xq cos a + ^0 sin a -p.
In the case when N^ lies between 0 and N we have to take
P-Po
OF THE STRAIGHT LINE 11
and when N, Nq lie on opposite sides of 0, ON^^ makes angle
(a + tt) with O'', and we have to take
In both these cases the length of the perpendicular is given by
- Xq cos a - y^ sin a + p.
(ii. ) If the equation of the line is given as
ax + bj/^c (r>0) . . . (1),
we have first to throw this into the " perpendicular " form.
►Suppose it becomes
X cos a + // sin a = 'p . . . (2).
Then, by equating the values we find from these two equa-
tions for the intercepts upon the axes, we obtain
cos a sin a p
a be
Therefore c cos a = ap,
c sin a = bp,
and c- = (a- + b^)p- ;
,-. c= \/(T- + b^^ p,
where there is no ambiguity in the square root, as both j' ^md c
are positive.
Hence cos a =
sin a =
b
c
and p — —r .,
^ 'J a? + b^
and the " perpendicular " form of the line
ax + by = c (O 0)
. ■ ax by c
vPTP "^ s/wTp ~ 'slWTW
Hence the length of the perpendicular from {x^, i/q) upon
ax + bi/ - c = 0
s/a^ + b'-
12 THE ANALYTICAL GEOMETEY
And the positive sign is taken when {x^, y^) is upon the opposite
side of the h'ne from the origin, the negative sign when it is on
the same side of the line as the origin.-'
This result holds for the equation of the straight line,
in Avhatever form it is given. The reason for the change of
sign in the expression for the length of the perpendicular is
that the equation of the first degree Ix + my + n = 0 divides the
plane of xy into two parts, in one of which Ix + my + n is positive :
and in the other it is negative. Upon the line the expression
vanishes.
Ex. 1. Tiausforni the equations
(i. ) 3.r ± 4?/ = 5 (ii, ) 3./; + 4?/ = - 5
into the perpendicular form, and from your tables write down tlie value of
a for each.
2. "Write down the length (if the iperpendicular from the origin upon the
line joining (2, 3) (6, 7).
3. Write down the length of the perpendicular from the point (2. 3) upon
the lines
\jc + iy^l, 5x+\2y = 2Q, 3.r + 4(/ = S.
4. P'ind the inscrilied and escribed centres of the triangle whose sides are
Zx + 4?/ = 0, 5x-12y = 0, y=\5,
and the equations of the internal and external bisectors of the angles of
this triangle, distinguishing the different lines.
[The student is referred for a fuller discussion of the subject matter of this
chapter to (i.) Briggs and Bryan's Elements of Co-ordinate Geometry, Part I.
chapters i.-x. ; (ii. ) Louey's Co-ordinate Geometry, chapters i.-vi. ; and (iii.)
to C. Smith's Elementary Treatise on Conic Sections, chapters i. and ii.
In all these hoolcs a large number of examjiles will be found illustrating
the points we have discussed.]
EXAMPLES ON CHAPTER I
1. Find the equation of the locus of the point P which moves so that
(i.) AP2+PB-^ = c2
(ii.) AP2-PB2 = c-^
(iii.) AP.PB = c-,
A and B being the points (-«, 0), {a, 0).
Ride. — To find the length of the perpendicular from a given point (o:„, y^)
upon a given straight line
lx-\-my-\-n = Q,
insert tlie values (,r„, ?/„) in place of {x, y) in the linear exjiression and divide by
the square root of the sum of the squares of the coefficients of x and y in this
expression.
OF THE STRAIGHT LINE 13
2. Find the equation of the straight line through ( - 1, 3), (3, 2), ami
show that it passes through (11, 0).
3. Show that the lines
3.>'- 2?/H-7--=0
ix+ 2/ + 3 = 0
19./;+ 13?/ =0
all pass through one point, and lind its co-ordinates.
4. Find the equations of the lines through the origin parallel and perpen-
dicular to the lines of Ex. 3 ; also those through the point (2, 2).
.5. Find the equation of the line joining the feet of the perpendiculars
from the origin u]ion tlie lines
4x+ il=\1
x + 2y = o.
6. Draw the lines
4?/ + 3a- = 12
Sy + ix=2i.
Find the equations of the bisectors of the angles between them, distinguish-
in£c the two lines.
7. The sides of a triangle are
,,:- ?/+ 1=0
X-ilf+ 7 = 0
x + 2ij-U=0.
Find (i. ) the co-ordinates of its angular points,
(ii.) the tangents of its angles,
(iii. ) the e(|uations of the internal and external bisectors of these
angles.
8. The angular points of a triangle are at (0, 0) (2, 4) ( - 6, 8). Find
(i.) the equations of the sides,
(ii.) the tangents of the angles,
(iii. ) the equations of the medians,
(iv.) the equations and lengths of the perpendiculars from the angular
points on the opposite sides,
(v.) the equations of the lines through the angular points parallel to
the opposite sides,
(vi.) the co-ordinates of the C.G.,
(vii.) the co-ordinates of the centres of the inscribed, circumscribed,
and nine-points circles.
CHAPTER II
THE MEANING OF DIFFERENTIATION
§ 11. The Idea of a Function.
If two variable quantities are related to one another in such
a way that to each value of the one corresponds a definite
value of the other, the one is said to be a function of the other.
The variables being x and i/, we express this by the equation
y =f{x) ; in Avhich case z and y are called the independent and
dependent variables respectively. Analytical Geometry furnishes
us with a representation of such functions of great use in the
experimental sciences. The variables are taken as the co-
ordinates of a point, and the curve, whose equation is
gives us a picture of the way in which the variables change.
So far as we are concerned in these chapters the equation
y -f{x) may be assumed to give us a curve. There are, how-
ever, some peculiar functions which cannot thus be represented.
v^ 12. Examples from Physics and Dynamics.
If a quantity of a perfect gas is contained in a cylinder
closed by a piston the volume of the gas Avill alter with the
pressure upon the piston. Boyle's Law expresses the relation-
.?hip between the pressure p upon unit area of the piston,
and the volume r of the gas, when the temperature remains
unaltered. This law is given by the equation
pv=]>^>\
0'
where p^, l■^^ are two corresponding values of the pressure and
14
THE MEANING OF DIFFERENTIATION 15
the volume. When the volume v for unit pressure is unity,
this equation becomes
2w= 1,
and the rectangular hyperbola, Avhose equation is
X1J = 1,
will show more clearly than any table of numerical values of p
and V the way in which these quantities change.
When the pressure is increased past a certain point Boyle's
Law ceases to hold, and the relation between p and v in such a
case is given by van der Waals's equation : — •
a and b being certain positive quantities which have been
determined by experiment for different gases. Inserting the
values of a and h for the gas under consideration, and drawing
the curve
with suitable scales for x and //, the Avay in which ^; and v vary
is made evident.
Such illustrations could be indefinitely multiplied. We add
only two, taken from the case of the motion of a particle in
a straight line.
When the velocity is constant, the distance s from a fixed point
in the line to the position of the particle at time t is given by
s = rt + Sq,
where s^ is the distance to the initial position of the particle,
and V is the constant velocity.
The straight line
y = vx + s^
then represents the relation between s and t.
When the acceleration is constant, the corresponding equation is
s = 1/^2 + ^_j^ ^ ,^^
where / = the acceleration,
Vq = the initial velocity,
5,5 = the distance to the initial position.
f
16 THE MEANING OF DIFFERENTIATION
In this case we have the parabola
y = h'^x- + r^x + Sy.
Also in both these cases we might obtain an approximate value
of s for a given value of /, or the value of / for a given value
of s, by simple measurements in the figures representing the
respective curves.
i^ 13. The Fundamental Problem of the Differential Calculus.
The aim of the Differential Calculus is the investigation of
the rate at which one variable quantity changes with regard to
another, when the change in the one depends upon the change
I in the other, and the magnitudes vary in a continuous manner,
y The element of time does not necessarily enter into the idea of
a rate, and we may be concerned with the rate at which the
pressure of a gas changes with the volume, or the length of a
metal rod with the temperature, or the temperature of a con-
ducting wire with the strength of the electric current along it,
or the boiling point of a liquid with the barometric pressure, or
I the velocity of a wave with the density of the medium, etc. etc.
The simplest cases of rates of change are, however, those in
* which time does enter, and we shall liegin our consideration of
the subject with such examples.
§ I i. Rectilinear Motion.
In elementary dynamics the velocity of a point, which is
moving uniformly, is defined as its rate of change of position,
and this is equal to the quotient obtained by dividing the
distance traversed in any period by the duration of the period,
the distance being expressed in terms of a unit of length, and
the period in terms of some unit of time.
When equal distances are covered in equal times this fraction
is a perfectly definite one and does not depend upon the time,
1)ut when the rate of change of position is gradually altering,
as, for instance, in the case of a body falling under gravity, the
value of such a fraction alters with the length of the time con-
sidei-ed. If, however, we note the distance travelled in difterent
intervals measured from the time t, such intervals beinir taken
smaller and smaller, we find that the values we obtain for what we
/
THE MP:ANING of DIFFEKENTIATION 17
might call the average velocity in these intervals are getting
nearer and nearer to a definite quantity.
For example, in the case of the body falling from rest we have
s=^hgf.
Let (s + &) be the distance which corresponds to the time
{f + 8f).
These quantities 8s and 8i added to s^ and f ave called the
" increments " of these variables.
Then .s + 8s - hg{t + 8t)" = hgt- + gt . 8t + hg(8ty',
8s
and .-. ~- -^ qt + }jg8f.
8t
It is clear that as 8t gets smaller and smaller, the " average
velocity " in the interval 8f approaches nearer and nearer to the
value gt. This value towards which the average velocity tends
as the interval diminishes is called the velocitij at the instant t, on
the understanding that wo can get an "average velocity" as
near this as we please by taking the interval sufficiently small.
The actual motion with these average velocities in the successive
intervals would be a closer and closer approximation to the con-
tinually changing motion in proportion to the minuteness of the
subdivisions of the time. The advantage of the method of the
Dirt'erential Calculus is that it gives us a means of getting
these "instantaneous velocities," or rates of change, at the time
considered, and that, when the mathematical formula connecting
the quantities is given, we can state what the rate of change of
the one is with regard to the other, without being dependent
upon an approximation obtained by a set of observations in
gradually diminishing intervals.
§ 15. Limits. Differential Coefficient.
If a variable which changes according to some law can be
made to approach some fixed constant value as nearly as we please,
but can never become exactly equal to it, the constant is called
the limit of the variable under these circumstances. Now if
this variable is .'•, and the limiting value of x is a, the dependent
variable y (where y^f{o')) may become more and more nearly
equal to some fixed constant value h as x tends to its limit '/,
18 THE MEAjS^ING OF DIFFERENTIATION
and we may be able to make y difter from h by as little as we
please, by making x get nearer and nearer to a. In this case
h is called the limit of the function as x approaches its limit a, or
more shortly, the limit of the function for x = a, and this is
written Lt^^,, (y) = h.
„ X . sin a;
E.g. {I.) It y=~r^
(li.) If /y =
1
or, more correctly, y has no limit for x = Q. '^
In this last example the function increases without limit as x
approaches its limit. AVe might have the corresponding case
of x increasing without limit and the function having a definite
limit : e.g. if
y = a^ where 0 < ft< 1,
M.= oo(y) = o.
This idea of a limit has already (§ 14) been employed, and
when s = hgt~, the velocity at the time t of the moving point is
what we here define as
In the general case of motion when the relation between
s and t is s —f(t), we take the distance at the time (/ + o^) as
(s + 8s), and we have
s + 8s^f{t + 8t),
8s _ /(/ + 8t) -fit)
8f~ 8t ■
Hence the velocity at the time t is given by
''-^^''=\8t)-^^"='\'~ 8t /•
" P^or a full discussion of the idi-a of /irnit, see Gibson's Calculus, chapter iv.
or
THE MEANING OF DIFFERENTIATION 19
This limiting value of the ratio of the increment of .s to the
increment of t as the increment of t approaches zero is called the
differential coefficient of s with regard to t. Instead of tc riling
Lt&t=J'v;], '^'<^ use the symbol —,,for this limiting value. It must,
\6t J ■ at
hoioever, he carefully noticed that in this symbol ds and dt cannot,
ds
so far as we are here concerned, be taken separately, and that -y,
stands for the result of a definite mathematical operation, viz. the
evaluation of the limiting value of the ratio of the corresponding
increments of s and t, as the increment of t gets smaller and smcdler.
We shall see later in § 38 that there is another notation in
which ds and dt are spoken of as separate quantities, but until
that section is reached, it will be Avell always to think of the
differential coefficient as the result of the operation we have just
described. It is clear that if 8t is very small, the corresponding
increment of s, namely 8s, will be very approximately given
by -7- • St. Still it is not a true statement, but only an approxi-
mation, to say that in this ease
This approximation may, however, be employed in finding
the change in the dependent variable due to a small change in
the independent variable, or the error in the evaluation of a
function due to a small error in the determination of the
variable, provided Ave know the differential coefficient of the
function.
We add some examples in which the differential coefficients
are to be obtained from the above definition, viz. —
If
^_f,f. ds_ ffit + 8t)-f(t)~^
Ex. 1. Us = at + b, -rr = a.
dt
1. If s = «<2 + 2W + c, ~ = 2(«< + h).
3. Ife = < '5^ = w.
at
4. lia; = «sinwc, -;- = aaj cos u/.
dt
20
THE iMEANING OF DIFFERENTIATION
i; IG. Geometrical Illustration of the Meaning of a
Differential Coefficient.
In the last sections we have been led to the idea of a limiting
value by the consideration of a moving particle, and have thus
been brought to define the
y
A
T
H
differential coefficient of s
with regard to /.
We have another illus-
tration of the meaning of
the differential coefficient
in the consideration of the
gradient, or slope, of the
curve
y
c
) K
t5.
4
Let P be the point (a:, //)
and Q the point (o: + &•,
V ~ %)) ^iitl let the tangent at P make an angle </> with Ox.
Then in Fig. 6
OM = .r
ON =x + &r'
^ and MP = // =/(.t) -\
MN = 8x
Thus the slope of the secant PQ
= tan HP(^)
^Sj -
8:r
_f(x + 8x)-f(x)
~ 8x ■ '
Now if we keep P fixed and let Q approach P, the secant PQ
gets nearer and nearer the tangent at P, and the limiting value
of the fi-action ^ as 8x gets smaller and smaller is tan c/j.
Thus, Avith the same notation as before
'^=U..i^\ = U
6a: =0
8x
Sx = 0
8x
tan (fy.
THE MEANING OF DIFFERENTIATION
21
Since the slope of the tangent is known when -y is found, we can
at once proceed to write dotvn the equation of the tangent at a 'point on
the curve y ^f(x), when the value of '- at that point is known.
Ex. 1. If/(a') = c""', write down /(.r + /i) ; and show that
Zt
fi.r)-f{x-hy
h
Interpret this result geometrically.
2. Find the value of -p at the point (2, 1) on the curve 4y = x^, and show
tliat the equation of the tangent at that point to this parahola is
^ 17. Approximate Graphical Determination of the Differ-
ential Coefficient.
When the equation connecting x and // is such that the curve
may be easily drawn, the slopes of the various positions of the
secants PQ, as Q is made to move nearer and nearer to P, will
give a series of values more and more nearly approximating to
the value of ^r ^t that point. An instructive example is the
case of the curve
in which the following table of values of 8x, 8// and can
ox
readily be obtained, and the way in which ; approaches its
ox
limiting value 2 at the point where x = 1 be made evident.
6a; jl
"J
•s
■7
■6
•5
•4 , -3 i -2 , -1
•09
•o,s
•07
■06
•00
•04
•03
•02
•01
Sy ,3
2-61
2-24
1-S9
1-56
1-25
•96
•69 -44
•21
•1881
•1664
•1449
•1236
•1025
•0816
•0609
•0404
•0201
Sx
2-9
2-S
2-7
2-6
2-5
2-4 2-3 2-2
2^1
2-09
2-08
2-07
2-06
2-05
2-04
2^03
2^02
2^01
22
THE MEANING OF DIFFERENTIATION
§ 18. In the chapters which immediately follow Ave shall
show how to obtain the differential coefficients of the most
important functions. This process of obtaining the differential
coefficient is called differentiating the function. We shall see
that in very many cases there is little difficulty in differentia-
tion, and that the knowledge of the differential coefficients is
of great value not only in geometry, but in the application of
mathematics to physics.
clv
From Fig. 7 it is obvious that when -f- is positive the
^ ax
tangent is inclined at an acute angle to the axis of x, and y
increases there with an increase of x, or decreases with a
decrease in .'•. When -f- is negative, the tangent is inclined
do' ^
at an obtuse angle to the axis of x, and ?/ decreases as x
fry
increases, or vice versa. When -^ = 0, the tangent is parallel
to this axis. Let us imagine the curve ABC to be a road,
and that a traveller is marching along it in the positive direction
of the axis of ,'•, which is horizontal. When the traveller ascends.
dy
'('/
j^ is positive ; when he descends, ~ is negative ; and if the
road is properly rounded off and no sharp corners occur, when
he passes from ascending to descending, or the reverse,
changes sign by passing through zero.
dy
dx
THE MEANING OF DIFFKEENTIATION 23
The acceleration of a moA'ing point is defined in Dynamics
as the rate of change of its velocity. Therefore, if we write
V for the velocity at time t, the acceleration at that instant is
dv
-J-.- If the position of the point at time / is given by 5; =f{t),
ds
then the velocity v
and the acceleration
dt
dc d /ds-^
dt^df' [dt
s =
^hgt
ds
dt~
--of,
dh
df-~
'9-
The differential coefficient of the differential coefficient is
called the second differential coeflticient, and in the case of s =f(f),
dh
is Avritten ^-7,.
dt-
E.a. If
and
EXAMPLES ON CHAPTER II
The differential coefficients required in the examples on this chapter are
to be obtained from the deiinition.
1. Plot the curves (i.) y = x + x'^
(ii.) y^x»,
and show that they have the same gradient when x = l.
2. By considering the area of a square and the volume of a cube, show
that the differential coefficients of x^ and x^ are 2x and ^x^ respectively.
3. Show that the curves y = x^ and y = x* intersect at the origin and
the points (1, !)(-!, 1), and that at each of the two latter points the angle
o
between the tangents is tan ~ '^ - •
4. Show that the gradient of the curve y = x^-'ix at the point where
x = 2 is 9. Find the equation of the tangeut there and trace the curve.
5. Find where the ordinate of the curve i/ = 3a;-4a:^ increases at the same
rate as the alascissa, and where it decreases five times as fast as the abscissa
increases.
6. U s = ut-lgf, fini^l the values of the velocity and acceleration at the
time t.
7. A cylinder has a height h ins. and a radius r ins. ; there is a possible
small error 5?* ins. in r. Find an approximate value of the possible error
in the computed volume.
24 THE MEANING OF DIFFERENTIATION
8. Find apjiroximately the error made in tlie volume of a sphere by
making a small error 5?- in the radius r. Tiie radius is said to lie 20 ins. ;
give approximate vahies of the errors made in the computed surface and
volume if there Ije an error of -1 in. in the length assigned to tiie radius.
Also calculate the ratio of the errors in the radius, the surface, aud the \-olume.
9. The area of a circular plate is expanding by lieat. When tlie radius
passes tlirough the value 2 ins. it is increasing at the rate of -01 in. per sec.
Show that the area is increasing at tlie rate of •047r sq. in. per sec. at that
time.
10. The length of a bar at temperature 0" is unity. At temperature t° its
length I is given by the equation
l = l+at + bt~,
find the rate at which the bar increases in length at temperature f, and give
an approximation to the increase in length due to a small rise in temperature.
11. If the diameter of a spherical soap-bubble increases uniformly at the
rate of "1 centimetre per second, show that the volume is increasing at the
rate of -277 cub. cent, per second when the diameter becomes 2 centimetres.
12. A ladder 24 feet long is leaning against a vertical wall. The foot of
the ladder is moved away from the wall, along the liorizontal surface of the
ground aud in a direction at right angles to the wall, at a uniform rate of 1
foot per second. Find the rate at which tlie top of the ladder is descending
on the wall, when the foot is 12 feet from the wall.
CHAPTER III
DIFFERENTIATION OF ALGEBRAIC FUNCTIONS; AND SOME
GENERAL THEOREMS ON DIFFERENTIATION
,5^ 19. The Differential Coefficient of r".
Let ij = x":
Then // + 5?/ = (a; + Sx)"-
Sx\ "
= a;" ( 1
dx
+ —
X
8//
X,
8x 8x
But by the Binomial Theorem, when h<\,
Therefore
(1 +/;.)» =.1 +;,//+ ^^-1 /,2 + . . .
„/, n ^ 71.72. - 1 (&j)"-' \
hy^_ \ X 1.2 ij;2 )
Sx &r '
Si/ , n.u - 1 ,,
. ■ . /^ nx" - 1 + -r--~ x>' - -8x + . . . ■■•-
8x 1.2
The fact that we have an iiitinite series on tlie right hand sometimes causes
difficulty to the student, as he imagines tliat what he calls the summing of the
infinite number of small terms involving 8:c, [dx)-, etc. . . . may give rise to a
finite sum. The answer to tliis difficulty in general is to lie fouml in a true view
of the meaning of a convergent infinite series, l>ut in the particular case of the
Binomial Series we are able to say what the possible error by stopping after a
25
26 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS
provided that 8x is so small that
X
Hence Lk,=o(^^^ =nx''-\
and the differential coefficient of x" is «x''~^
This is true whatever value n may have, provided it is in-
dependent of X.
Thus ^('^)-A^S
dx
tx.
dx \x<J
i
X'
2
3 \/.7:
§ 20. General Theorems on Diflferentiation.
Before proceeding to obtain the differential coefficients of
other functions, it will be useful to show that many complicated
expressions can be differentiated by means of this result, with
the help of the following general theorems : —
Proposition I. Differentiation of a Constant.
It is clear that, '\i y - a, the slope of the line is zero, and
dv
~ = 0. In other Avords, it is obvious that if a magnitude remains
the same its rate of change is zero.
Thus the differential coefficient of a constant is zero.
Proposition II. Differentiation of the Prodiict of a Constant and
a Function of x.
Let y = au, where (/ is a constant, and u is a function of x.
When X becomes .'■ + ax, let u become u + 8u, and y become
y + 8y.
certain inuuber of terms can be, and thus exclude the infinite series from onr
argument.
It is, liowever, worthy of note that the formula for the differential coefficient of
X" can be obtained without this series, by taking first of all n a positive integer
and then using § 20 Prop. VT.
GENERAL THEOEEMS ON DIFFERENTIATION 27
Then y + Sy — a{u + 811),
- 8y 8u
and ^ = a— •
8x 8x
Therefore Ltsx=o\j-,J = <^Ltsx=o ( v-, j ,
dy du
or -f- = a-j--
dx dx
.'. The differential coefficient of the ^^Toduct of a constant and a
function is equal to the product of the constant and the differential
coefficient of the function.
The geometrical meaning of this theorem is that if all the
ordinates of a curve are increased in the same ratio, the slope of
the curve is increased in the same ratio.
Proposition III. Differentiation of a Sura.
Let y = ";/ + r.
Then, as before, y + 8y = {u + 8u) + (r + 8v),
8y 8u 8v
8x 8.r 8x
Proceeding to the limit,
dy du dr
dx dx dx
The same argument applies to the sum (or difference) of
several functions, and we see that the differenticd coefficient of such
a sum is the sum of the several differential coefficients.
Ex. Differentiate the following functions : —
(i.) x{2 + ,-f
(ii.) {a + b.>: + cx-)\'x
,... , x^ .7'3 ,->;2 X , \ \ 1
2 + 2x+Zx'
(iv.)
\x
Proposition IV. Differentiation of the Product of Tv:o Fvnctions.
Let y — uv,
Then, as before, y + 8y = {u + 8u)(v + 8v).
28 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS
Thus hy - vSu + uSr + 8u . 8v,
■J St/ 8u 8v „ 8v
and ^i = v +u■^ + 8^l.^■
bx 8x bx 8x
Proceeding to the limit,
dy d'U dv _
dx dx dx '
since as 8x approaches its limiting value zero, 8u approaches
, , 8u 8v - du , dv
zero also, and ^, ^^^ become^- and -5--
bx 8x dx dx
This result may l^e written
1 dy 1 du 1 dv
y dx u dx V dx
and when y — uvw, we Avould obtain in the same Avay,
1 dy 1 du 1 dv 1 dw
~ ■ ^ = - • :r + ~ ■ T + - ■ ^- (Cf. ^31.)
y dx u dx V dx w dx
In the case of two functions it is easy to remember that the differ-
ential coefficient of the product of ttvo functions is equal to the first
function x the differential coefficient of the second + the second function x
the differential coefficient of the first.
Ex. Differentiate the following functions : —
(i.) (l+ic2)(2a;'-^-l)
(ii.) (2a;2+l)(a; + 2)2
(iii.) iax + bf{cx + df-
(iv.) x{x + l){x + 2),
and show that the results are the same if the expressions be multiplied
out and then differentiated.
Proposition V. Differentiation of a Quotient.
Let y = ujv.
u + 8u
Then y + 8y =
V + 8v
u. + 8u u vSu - uSv
and ^^^v
V + bo V , / , bV
L-\\ +
V V
GENERAL THEOREMS ON DIFFERENTIATION 29
8ii
8r
Therefore
8>/ _
8x
8x
■!)
Proceeding
to the
limit,
it follows that
dm
dv
dy
'%■
dX .;,
dx
V^
In words, to find the differential coefficient of a quotient, froiu
the product of the denomincdor and the differential coefficient of the
numerator subtract the product of the numerator and the differential
coefficient of the denominator, and divide the result bi/ the squa;re of
the denominator.
Ex. Differentiate the following expressions : —
(i-)l^ (ii.) ^^'^^^'^ (iii.) "■
(a; + 3) ' '' (a; + l)(a; + 2)
These five formulae, with the help of the result of § I'J.
enable us to differentiate a large number of expressions, but
they do not apply directly to such cases as Va + x, s/a,'^ x^,
1 1 '
ax + b {ax + by
Each of the above expressions is a function of a function of
X, and we proceed to prove the general theorem :—
Proposition VI. Differentiation of a Function of a Function.
Let // = F(m)
where u =f{x)
f'.g. 1/ = \ u
where u = a~ + x-
Then when x is changed to ;/: + 8x,
let u become n + 8u,
and y become // + 8y ;
* Tlii.s result may be obtained by writing
and then differentiating both sides of this equation.
30 DIFFERENTIATION OF ALGEBRAIC FUNCTIONS
the functions being such that for a small change in r, Ave have a
definite and small change both in u and i/.
-n, j_ 81/ 8y Su
But ^ = ^ • ^>
dx 6u 8x
.-. proceeding to the limit,
Thus djj^dy^(lu^
dx du dx
Ex. 1. When y = ___^,, we may put
and u = x + a)
1^
dy "\?i-/ du
'<.i)
, — , - , -, where w = a; + a,
dx du dx
= -4.1
{x + af
2. When 2/ = (a^ + ^)", prove -- = ?ia(aa; + &)»-i.
3. When ?y = (1 - x) \/r+^2
g=(l-.)|(^/^^w^:^^(l-.).
•D 4. '^ /, ., d\lu du , ^ .,
but -3- rjl+x- — —, — • -;— , where u = l+ x^,
dx du dx
= (~^)i2x)
V2\/l+a;V'
>Jl+x^
di/ _ (1 - x)x .
_ -Ix'-x+l
. ^,71. /ax+b dy ad -be
4. When y= ^ ^^^, prove ^=2V(ax4-fc)(c.. + rff
GENERAL THEOREMS ON DIFFERENTIATION 31
EXAMPLES ON CHAPTER III
Find V^ in the followinc' cases :-
(Ix
(i
/=(,v'x-^'-)" (vii.)^ = ,
Cb + ft.!-'"
(ii.) 2/= n/2^^:^^^P (viii-) 2/=(l+.r")'
(iii. ) ?/ = \/(.r + 1 )(a' + 2) (ix. ) y = N-'x-^ + a2 + s V^ - a'-^
(iv. ) y = {x + a)H,r. + by (x. ) 7/ = -,-== + ,- :^
\' X' + a- \ix- - a''
/l+x , . . a^
, . , {a-x)p , .. , /T+x + a?
2. Find the gradient at the point (./„, 2/„) in the following curves : —
(i.) y- = 4ax
(ii.) x- + y' = a-
(iii.)j±i:^i
(iv.) 2j'2/ = c-.
3. Prove that the equations of the tangents at {Xq, */„) to these curves are
respectively
(i-) 2/2/0 = 2«(.'' + a-o).
(ii.) xXn + yijn^aA
,iu.,?±t=>.
(iv.) xy^ + y.r^^c-.
4. A boy is running on a horizontal plane in a straight line towards the
base of a tower 50 yards high. How fast is he approacliing the top, when he
is 500 yards from the foot, and he is running at 8 miles per hour ?
5. A light is 4 yards above and directly over a straight horizontal patli on
which a man six feet high is walking, at a sjieed of 4 miles per hour, away
from the light.
Find (i.) The velocity of the end of his shadow ;
(ii.) The rate at which his shadow is increasing in length.
6. A man standing on a wharf is drawing in the painter of a boat at the
rate of 4 feet per second. If his hands are 6 feet above the bow of the boat,
prove that the boat is moving at the rate of 5 feet per second when it is 8
feet from the wharf.
7. A vessel is anchored in 10 fathoms of water, and the cable passes over a
sheave in the bowsprit which is 12 feet above the water. If the cable is hauled
in at the rate of 1 foot per second, prove that the vessel is moving through
the water at a rate of 1^ feet per second when there are 20 fathoms of cable
outt
^tra:
UNIVERSITY )
OF
32 DIFFEEENTIATION OF ALGEBRAIC FUNCTIONvS
8. If a volume r of a gas, contained in a vessel luitler pressure jr;, is com-
pressed or expanded without loss of heat, the law connecting the pressure and
volume is given by the formula
y^v''^ constant,
where 7 is a constant.
Find the rate at which the i)ressure changes with the volume.
7 .2
9. In Boyle's Law. where iiv — c-, show that -r-= — .,. What does the
' dp p'
negative sign in this exjuession mean ?
10. In van der Waals's equation
I ;;+„]('' - //) — constant.
Prove that
dv _ {v - h)
dp ( a 2ab\
CHAPTEE IV
THE DIFFERENTIATION OF THE TRIGONOMETRIC FUNCTIONS
{The angles are supposed to he measured in Radians)
§ 21. The Differential Coefficient of the Sine.
Let y = sin x.
Then y + % = sin (.'■ + 6x),
and S'/ = sill (.'■ + ?>:<■) - sin x
8x
= 2 cos ( ;r + — ) sin
/ 8x
8y ( ^4^^
Therefore &7. = ^°H''' " o") 1 "l|r"
Proceeding to the limit, and remembering that
/sin d'
Lt.A-^ ) = 1, it follows that
dil
-f = cos .'•.
(I.r
K.li. — When y = sin iinx + u)
dii dii da .
-^ -^ • -T- where u = mx + v
dx dii dx
d(sm u) du
d\L dx
- cos ?« . m
= m cos {mx + ii).
Ex. Prove from the detinitiou of —, that when 7/ = sin {mx + n),
CLdb
(III , .
-^=vi COS {m.y + n).
ax
4 33
34 THE DIFFERENTIATION OF THE
§ 22. The Differential Coeflacient of the Cosine.
Let y = cos X.
Then y + % = cos (;*• + 8x)
and 8y = cos (.f + 8x) - cos x
■ ( h\ . &
= - 2 sin \x. + -J sm -.
i
2
Proceeding to the limit,
di/
, = - sin z.
ax
k
N.B. — When y = cos {mx + n), -y - -in sin (i/w; + n)
Ex. Prove from the definition of -f-, that wlien y — cos, {iax+ it),
-3-= -msm {mx + n).
§ 23. The Differential Coefficient of the Tangent.
_ sin z
Let y = tan x =
cos a:;
c/(sni x) . d(cos x)
n., 7 cos x ^—^ sin X —
Then dy _ dx dx
dx
cos'^x
cos^^ + sin^;i;
cos^.?:
1
cos-:i;
^z
sec^x.
N.B. — When y = tan (mx + n), j-. =" ""^ sec-('mrc /()•
Ex. Prove from tlie definition that when y — tAii {mx + n),
-rr- - 111 sec- iinx + n).
ax
TRIGONOMETRIC FUNCTIONS 35
From these three results it is easy to deduce the following : —
— - . cot .*■ = - cosec-.r : -r- ■ cot hm: + n) = - m cosec-(m.r + n)
ax ax
I sin z d , , sin {mx + n)
sec ;'; = — 5- ; — • sec \inx + n) = in
dx ' cos'' r ' (/.'■ ' co^\mx + n)
d cosx d . . cos (mx + n)
-r • COSeC X= r-TT ', ^^ ' COSCC (lltX + '11)- - 111 . ., ^.
dx sm^x' dx ^ ' sm-{mx+n)
§ 24. Geometrical Proofs of these Theorems.
All these cases of differentiation may be discussed geometri-
cally. The method will be followed
easily from the case of the tangent,
which we now examine.
Let Z.MOP be the angle 0 radians,
and let OM be 1 unit in length.
Let zlPOQ be 86, and let QPM be
perpendicular to the line OM from
which 6 is measured.
Let PN be perpendicular to OQ.
Then 8(tan 0) = PC^
= PN sec ZINPQ
= PN sec {$ + 86)
= OP sec (6 + 86) sin 86
- sec 6 sec (6 + 86) sin 86.
rvv. 8{iiin6) /n , sa\/^^^\
Thus ^-^ — ^ = sec ^ sec (^ + S^)f — r^ j,
and proceeding to the limit,
c/(tau 6) ,^,
-d6 ='"''^-
Examples. Find —- in the following cases : —
(i.) y = 2a sin {hx + c) sin [hx - c).
(ii. ) y = X' cos 2x.
(iii. ) y = tan 3a; + cot 3a,'.
, . , sin 2x - sin x
(iv. ) y= •
'^ cos a;
(v.) y = x"^sm'^x.
(vi.) y = x"^ sin 7ix.
(vii.) ?/ = sin-^ aj COS' a'.
(viii. ) y = sec- {ax +b) + cosec- (ex + d).
36 THE DIFFERENTIATION OF THE
§ 25. The Inverse Trigonometrical Functions.
Since the sine of an angle varies continuously from - 1 to +1
as the angle passes from - - through zero to + ^ , it is convenient
to take the inverse sine as lying in these two quadrants. In
other words, for
^ = sin " ^,'
we take that part of the curve
sin // = a', ■
which lies between // = - -^ and y = -^ -■
In this case, when
• 1 / ^ ^\
?/ = sm \r V o^.'^<2/
sin y = a;
and difterentiating, ,-(sin //) = -r (■'')■
ax ' (l.r
„, , (/(sin y) ihf ^
Iherefore , •^' .-^=1
ciy ax
or
But
and therefore
XT ^
Hence ^(om .. , - , —
do-> ' + v'l -,r-
§ 26. The DiflFerentiation of the Inverse Cosine.
In the case of // = cos~\'-,
it is convenient to take // as lying between <» and tt, and in this
case the equation
cos ;// •
(l.r
C0S;^ =
+ ^'l
- .'■-, since -
n TT
dy
1
dx "
+ x/r
- x'^
,-r,A -
1
on difterentiation, gives
d dy
^{cosy) . ^ = 1,
di/ d.i
TRIGONOMETRIC FUNCTIONS 3 7
d;,
(0 <//<-)
or,
- sin
d.r
and since
sin // = +
s'l-x'
di^ _
1
dx
x/1-7^'
or
J-(cos-V)= -
1
/i To
This result may obviously be derived from that for sin"^r,
since sin"^,*: + cos~^,'' = -•
2
^'21. The DiflFerentiation of the Inverse Tangent.
In the case of the inverse tangent we get a complete set of
values by taking // in the interval - ~ to + -•
When // = tan " ^x
tan // = ,'■,
and differentiating,
(/(tan ij) d//
dji dx '
or sec-// • ,- = 1.
dx
Hence (tan~^.r) =
dx 1 + .''-
If the student Avill examine the graphs of the functions
sin-^r, cos "I'', etc., he will see that without the above restrictions
on the size of the angle there would be an ambiguity in the
results for the sine, cosine, secant, and cosecant. For a given
value of .r, Avithin the possible range of values for .'■, Ave have an
infinite number of values of //, and at these points on the curve
the gradients are equal in magnitude, but may be opposite in sign.
Ex. Write down the values of ■ in the following cases : —
dx
(i.) 2/ = sin-M -j+cos-'(^j. (iv.) y=tan-i(^j— ;
(ii.) y — s,m.-'^{\-x). (v.) y = x^fim-'^{\>r).
(iii.) ?/ = cos-M — '^,\. (vi.) ;/ = tan-H''''')-
38 THE DIFFERENTIATION OF THE
EXAMPLES ON CHAPTER IV
1. Differentiate the following functions : —
(i.) sin-'x + cos'',''.
(ii. ) tan ;>• + - tan" ,vj.
o
(iii.) sec- a; + tan- :)'.
(iv. cosec- X + cot- o:
1 + sin X
(v.)
VI.
1 - sin X
1 - cos X
1 + cos X
, Ti- sm a dii cos^ a; - sm" a;
2. It */ = :, — , prove that -r- = , . -„.
1 + tan X ^ ax (cos x + sm xy
S. If ?/ = cos (.'-■'), prove that ~= - 3x- sin (a/), and find ^ when
ax ^ ax
(i.) 2/=a;™sina-".
(ii.) y = x''"'Cosx".
(iii.) 2/ = a:™ tana".
4^. Differentiate the following functions : —
(i.) (a!2 + l) tan-la; -a;,
(ii. ) X .sin-i x + \/T^^.
(iii.) tan-'( ^''"'"^ ) . . (Put \'.r= tan 0, v'« = taiia.)
(iv. ) tan-''
^-X-rX'
(V.)
cot-i(l±yLL^) . . (Put a- = tan e.)
5. A particle P is revolving with constant angular velocity w in a circle of
radius a. The line PM is drawn from P perpendicular to the line from the
centre to the initial position of the particle. Find tlie velocity and accelera-
tion of M.
6. If the position of a point is given at time t by the equations
a; = a(ct;^ + sin w<),
y = a{l - cos ut),
where a and w are constants, find its component velocities and accelerations,
and its direction of motion at the time t.
7. Prove that when
ax 3. y/,,.^ _ I
and that when
^ 1 ^ / 1 ^ 1
dx xsix^-l'
and illustrate your results from tlie graph of the inverse secant.
TRIGONOMETRIC FUNCTIONS 39
8. Prove that when
^>'^r ^^ (cosec-' X) :-
die xfja^ — l'
and that when
a;< - 1, -y- (cosec"' x)—
dx ■ .vsV-i'
and illustrate your results from the graph of the inverse cosecant.
CHAPTEE V
THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS — PARTIAL
DIFFERENTIAL COEFFICIENTS — DIFFERENTIALS
§ 28. In this chapter we assume a knowledge of the properties
of the following series : —
e« = 1 + .!• + — - + ...
A,
a* = 1 + ,'■ log « + 1^ (log of + . . .
which hold for all values of .'■, and
fi .7-3
log(l+.0 = :i--^ +-- . . .
which holds Avhen - 1<.'< + 1, using "log .':" for "logg.''."
We shall now show how to differentiate e^, «^, log ;'', and other
functions whose differential coefficients may thus be obtained.
§29.
To differentiate e*.
Let
]) = e""-
Then
y + 8y = e^+^^,
.-. Sij = e''+^' -(.'■'' ,
= e'(e^^- 1),
J, 8x (SxY
5y w, «■>; (&)-
40
EXPONENTIAL AND LOGARITHMIC FUNCTIONS 41
Proceeding to the limit,
dy
dx
d^.
Thus
i'-)
= e^.
It follows
that the
equation
dy _
ilx
■-y
is satisfied by // = ce^, where c is any quantity independent of x.
p]x. 1. If v/ = «e''-^, prove that '-^ = />//.
■2. It u-e'-, prove that j-^=-2,nj.
3. Uy = .t-e*'", prove tliat .'■ y-( = ( 1 4- 2.>-)?/.
4. Prove, /ro7« the definition of the differential coefficient, that if \j — ae^'^,
aa;
I 30.
To differentiate log '.
Let
y = log ,'■.
Then
y + 8y = log (x + dx),
and
Sy = log (x + d/;) - log .'■,
= 7-2(7) ^aU -■ •
• ^K?)<^-
6//_ 1 1 /6x\ 1 /6,r
Proceeding to the limit,
dy^l
dx X
Ex. 1. If?/ = «^, log v/ = .f log «. Hence show that
(fa '
— {a^) = a^Aogea.
2. It 2/ = log„ .1; prove that -^ = — =^^ = — ,
d'- ,'• .'■ . log,, a
3. If 2/ = log («. + &), prove that 1=^^.
42 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
4. Prove the result of (3) from the definition of the differential coefficient.
5. If ^ = log -y , prove that -r = ^^— w, — ^•
\o- X ) ^ rlx [a - X) {b - x)
6. If 7/ = log {x+ ^x~±a-), prove that -j- = — = .
^ tj. , ta-b cos Q , , ^ dy ab sin 6
I. If w = lo£f . / 5 T , prove that --^ = „ ,„ „„■
^ ^ V a + b cose' '■ dd a^-b- cos^ 6
v^ 31. Logarithmic Differentiation.
We have already obtained a general rule for the differentia-
tion of a product or quotient. We are now able to prove
another method which often leads more quickly to the result.
This method is called Loi/arifhmk Differenfidfion.
Let y/= nnc.
Then log y - log n + log r + log tr,
d ,- , dy d ,. . du d ,. , dv
d dw
d2o dx'
1 dy 1 du 1 dv 1 die
y dx u dx V dx w dx
In other words, before differentiation of an exjjression involving the
irrodud or quotient or poioers of other expressions, take logarithms of
both sides of the given equation.
T- -, 1,- {ax+bY{cx + dy' 1 dy an qr se
Ex. 1. It // = ' - ■ • - ' ' ^
{ex+fY y dx ax + b cx + d cx-\-f
2x
2 If,,- ./^±^ ^- /
^- 112/- V i_^,... ,fo.- ^/(l+:^^2)(l_^2)a-
^ _ /(i. + 2bx + cx'^ ^^y _ b{a-cx')
V (I - 'Ibx + ex- dx (a -
{a - 2bx + ca;2)»(a + 2bx + cx'')i
S 32. Important Example.
If y = c-' "■' sin hx,
dv ■ ■, d , ^ . „>. ^ / • 7 X
-f- = sni bx -r (e~'^^) + e~""^ ■ -y- (sin bo-)
dx dx dx '
= e ""•'"(- a .sin bx + b cos bx).
PARTIAL DIFFERENTIAL COEFFICIENTS 43
NoAv if a = tan ~^{^), o. and b being positive,
a
cos a =
Sin a =
b
/7y/
and -r- = - x^ft"-^ + b- . e~^^ (sin fo cos a - cos bx . sin a),
= - v^'?' + b- . e'"-^ sin (fe - a).
Thus the tangent to y = r-""^ sin fo' is parallel to the axis of x
when bx = n-n- + a,
and the equation defines an oscillating curve with continually
diminishing amplitude in the waves as we proceed along Ox.
It is easy to show that when
y - gaX gj,;^ Qjj. _L ,.^^
-1 = v/o^T- f"^ sin (bx + c + a),
ax
and that here the waves increase in amplitude ; and corre-
sponding results hold for the case of the cosine.
v? 33. Maxima and Minima Values of a Function of one
Variable.
The student is already familiar with the graphical and
aloebraical discussion of the maxima and minima of certain
simple algebraical expressions. The methods of the Differential
Calculus are well adapted to the solution of such problems,
since, if the graph of the function is supposed drawn, the
turning -points, or places where the ordinate changes from
increasing to decreasing, or vice versa, can only occur where
the tangent is parallel to the axis of x, as in the points
Ap A2 . . . of Fig 9, or where it is parallel to the axis of y as
in the points B^, B., . . ., except in such cases as the points
Cp C.3 . . ., where, although the curve is continuous, the gradient
44 EXPONENTIAL AND LOGAEITHMIC FUNCTIONS
suddenly changes sign, without passing through the value zero
or becoming infinitely great.
In case (A) : . is zero at the turning-point; and if this point
is one at Avhich the curve ceases to ascend and begins to descend,
dy
dx
- changes from being positive just before that point to being
Fici. 9.
negative just after. At such a point the function is said to
have a maximum value. In the other case, where the curve
ceases to descend and begins to ascend, y- changes from nega-
tive to positive, and we have a minimum. In Fig. 9, at A^
there is a inaximnm ; at A., there is a minimum.
In case B : -r_ is, infinitely great at the turning-point, and
ftJy
at Bj, where there is a viaxim,um, it changes from positive to
negative, while at Bo, where there is a minimum, it changes from
negative to positive.
The other turning-points Cj, C^ in Fig. 9 correspond to dis-
PARTIAL DIFFERENTIAL COEFFICIENTS 45
continuities in ~, but it can be shown that these will not
a,r.
occur in the functions with which we are dealing.
.^34. Points of Inflection.
Althoue;h the vanishini*; of ^ is a necessari/ condition for a
'^^ '^ dx
maximum or minimum, it is not a sufficient condition, since
the gradient of the curve may become zero Avithout changing
its sign as we pass through that point. Examples of such
points are to be found in Dj, D.^ of Fig. 9. In the case of I)^
the gradient is positive before and after the zero A^alue ; in the
case of D^ it is negative. At these points the curve crosses its
tangent, and when this occurs, whethei' the tangent is horizontal
or not, the point is called a point of inflection.
We cannot here discuss the analytical conditions for such a
point in general ; but in the cases of horizontal tangent (case D)
we see that -— vanishes and does not change sign ; and in the case
if,
of vertical tangent (case E), y^ is infinitely great at the point,
and does not change sign as we pass through it.
Ex. 1. Show that ij = a.i'- + 2h.i' + c has always one turning-point; and
point out when it is a maximum and when it is a uiinimuni.
2. Find the maximum and nunimum ordinates of tlie curve // = ;r^ - 6.'''-+ 12,
and also find the points of maximum gradient.
3. Find the turning-points of the curve 1/ =(;>• + !)■'(,'•- 2 j'', and show that
( - 1, 0) is a point of inflection.
(,v-l)
4. Find the turning-points ot y = -rr, — ^/
§ 35. Partial Differentiation.
So far we have been considering functions of only one
independent variable, y =/(•'). Cases occur in Geometry and
in all the applications of the Calculus where the quantities
which vary depend upon more than one variable. For instance,
in (xeometry the co-ordinates of any point (.r, ?/, :■) upon the
sphere of radius a, whose centre is at the origin, satisfy the
relation .'■- -f //- -f :- = a'.
Hence we have z' = a- - .'- - //"-,
46 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
and if we cut the sphere by a plane parallel to the //: plane,
along the circle where this plane cuts the sphere x is constant
and the change in z is due to a change in // only. In the
section by a plane parallel to the zx jDlane, the change in z
would be due to a change in .'; only. Similar results hold for
other surfaces.
Again, the area of a rectangle whose sides are x in. and // in.
is xji sq. in., and we may imagine the sides x and y to change in
length independently of each other ; while the volume of a
rectangular box whose edges are x, y, and ,:; in. is o-yz cub. in.,
and x, y, z may be supposed to change here independently.
The ordinary gas equation
pv
jt;^ = constant
is another example of the same sort of relation, and it would
be easy to multiply these instances indefinitely.
§ 36. Let the equation
express such a relation between two independent variables .'• and
v/, and a dependent variable „.
Let us suppose that the independent variable y is kept
constant, and that ./■ changes.
Then the rate at which z changes with regard to ,/•, when //
is kept constant, will be given by
(f(x + 8x,y)-f(x,y)]
In the second case let ;/; be kept constant and let // change.
Then the rate at which z changes in this case is
^fh = Oy ^^ }■
These two differential coefficients are called the Partial
Differential Coefficients of ~; with regard to x and y respectively,
7)~ Pi
and are written ^ and ^ respectively.*
ox dy
'>
cz
* It is hardly necessary to point that this symbol 5— stands for an operation,
and that ds, dj/ are not to lie considered separately ; also that this is a different
notation from the d.c of our earlier work.
PARTIAL DIFFERENTIAL COEFFICIENTS 47
Oz oz
Ex. 1. When z = .a/, prove from the definition that ^ = v, and t^^ — x.
' oa; dy
2. When 2as = a;- + y-, i)rove from the definition that ^^ = -, and 7^^ = ' .
3. If ii — jnjz, prove from the definition that— = ?/c.
§ 37. Total Differentiation.
When the variables x and // in the above examples Ijoth
depend upon a third variable t, say, ,:: will vary in value as ,/• and
// change with t.
In the case z = xy
z + Si' = (,/• + 8./:)(v/ + 8//)
and ^z 8x 8// 8x
8t^^^ ¥"-■'¥ "-81 ■"'
so that, proceeding to the limit,
dz dx dii
dt ' dt dt
But dz , dz ,
y = 7r and X ~ ~ when ~ = xy,
therefore, in this case
(h dz dx dz dy
7t~ dj-' dt^dji ' Tif '
In the second example,
2az = X- + ,//-,
we find
2a& = %i'8x + ■2y8ij + {8xf 4
and
dz ' dx dy
''dt=-'dJ'-^df-
so that again
dz dz dx dz dy
di~di.' dt^ dii' dt
It can be shown that this holds in genei^al, but the proof of the
theorem cannot be taken at this stage of our work.
The differential coefficient ,^ is called the Total Differential
dt
Coefficient in such cases, as compared with the Partial Differen-
tial Coefficient defined above.
48 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
As a special case, when z =/('■, y) and v/ is a function of ,'■, we
obtain
dz dz dz cly
d.r dx dy dx
and the left-hand side is called the Total Differential Coefficient
of z with regard to x.
Also the result that when z =f(x, y) and ,'-, y are functions of t,
dz dz dx d: dy
df ~ dx df "^ dy dt
may be used to obtain an approximation to the small change 6~
in z due to the small changes 8x and ?>y in ,'■ and //, when t
becomes t + 8f.
For, as we have seen already (p. 19),
dx
di
dy
Jt
dz
dt
and we thus have, on multiplying the above equation by M,
8x will be approximately
8z
8t,
Si,
8t
8z = ^- 8x + ^^ ■ 8//.
dx
38. Differentials.*
r)y
In the case of the curve
y=f{x) the increment 8y of y
which corresponds to the in-
crement 8x of .'■ is given in
Fig. 10 by HQ.
Also
HQ = HT -f TQ = 8x ■ '^ + TQ,
(IX
. • . 8/1 = 8x
dy
dx
TQ.
M
Fk;. in
N
As 8x ffets smaller and
smaller, TQ gets smaller and
smaller, at least in the neigh-
* § 38 may be omitted on first iviidiiig.
PARTIAL DIFFERENTIAL COEFFICIENTS 49
boiirhood of P ; and the " small quantity " TQ is a smallei-
" small quantity " than S.i; since
Sy _ dy T(^)
1 . 1 ,. . S'/ . , (hi , TU ..
and ni the limit — is equal to ,, so that — must disappear in
&r ^ (It Ac ^^
the limit. In mathematical language, if 8x is an infinitesimal (or
small quantity) of the first order, TQ will be at least an infini-
tesimal of the second order.
It is convenient to ha^e a name and syml;»ol for this quantity
dy
tr
symbol is " dy."
Hence with this dehnition of the term " differential,"
Y ■ Sx. The name adopted is the "differential of y," and the
where we have enclosed ~ in l)rackets on the right-hand side
d.r
SO that it may Ite clear that this stands for the differential
coefhcient ol)tained by the processes we have been developing
in the preceding pages.
By the above definition
'^(/(■'■))=/V)-S-', where ,/>) = ':^^
and dx = 8x.
So that dy =/'(•'■) • '^'''i when // =/(■'■)•
Hence we may restate our definition as follows : —
The differential of fhe independent variable is the actual increment
of th(d variable.
The differenticd of a function i>> the differential coefficient of tlic
function multiplied by the differential of the independent variable.
In this definition it is not necessary to assume that the
differentials are small quantities or infinitesimals, but in all the
applications of this notation this assumption is made. In that
case the equation
dy=f'{x)dx
5
50 EXPONENTIAL AND LOGAKITHMIC FUNCTIONS
will give the increment of y, if small quantities of the second
order be neglected.
Such an equation as
dy =f{x)cb;
a differential equation as it is called, may be used in this
way to give the approximate change in the dependent variable,
and from this point of view it saves the trouble of writing down
the equation between the increments, and then cutting out the
terms whose smallness is such that they may be neglected.
Ex. 1. Write down a table of differeutials corresponding to the standard
differential coefficients.
e.g. d{x") = )i:e"-''dx.
du
2. If a,' = a cos 6, y = a sin 6, prove by differentials that -^= - cot 0.
.-> Tf I X ■ ^s /I ^\ ii i dy sin ojt
6. It x = a[M + sixi ut), t/ = a[l -cos wt), prove that -f- = ]
dx 1 + cos ut
4. If ~ = xij, prove that dz — ~dx + ~ dy.
cx cy
EXAMPLES ON CHAPTER V
1. Find the differential coefficients of
(i.).™-^, (ii. ) a;'»e»^, (iii.) (aa;- + 6)c''^+'', (iv.) c^«"~^^.
2. Find the differential coefficients of
(i.)ci+-^", (ii.) a-V^", (iii.) ./-'"C"", (iv.) .f'"a^".
3. Find the differential coefficients of
{\.)x-nogx, (ii.)log(^'^^), (iii.)log(v'.^^+V-^l), (iv.) log (^^^j,
4. Differentiate the following expressions logarithmically : —
(i.) n/(2.^ + 1)(,«-2), (ii.) v^+^.' (iii-) ^2(^)3. (iv.)^-,
^ ' cos'^wa;' ' \ x)
and point out why we cannot apply our formula for the differential co-
efficient of a;" to the case of ,r^.
1 f ax + b \ ., , di/ 1 , ,„ ,
5. I{v= I tan~M , , prove that ~= — 5 — --, (acyb^.)
. ,, 1 , {x + lf 1 ^ , /2,c-l\ ^, ^dy 1
PARTIAL DIFFERENTIAL COEFFICIENTS 51
7. If 2/ = 2 cos-i V ^^, pi-ove that ^= ^^ ^ _ . (a > a- > ^.)
8. If 2/ = ,1_ cos-J ^ Z'' r_^', prove that 5^= ^=^ . (cc < a < ,S.)
^ T„ , /b + atiosx+ sjb- - ofi sin x\ ^i ^dy Jb'^ - a^
9. If w = log ( T , prove that ^ = -—^ .
\ a + beosx J ax a + b cos x
10. it / -— — -== tan-i{ A / , tan - ,-, prove that -ts = — -, -. •
sja- -b- l^ a + b 2 J ^ dd a+b cos d
{a^>b-\)
11. Ill the curves whose equations in polar co-ordinates are (i.) )- = ae9cota,
(ii.) r"-a" sin nd, (iii.) r" = a" cos n6, (iv.) ?•" = «« sec 7id, (v.) r" = «" cosec«^,
find r-T- • Can you give any geometrical meaning to this expression ?
12. If 2/ = e-'-^sin(2.r+l), prove that -^ = 2 ^''2 • e-^-^ cos (2./-+1 -f ^).
13. Find the value of V in the following curves ; discuss the way in which
dx
it changes as x passes along the axis ; and find the turning-points, if there
are any, of each curve ; —
{i.)y = x(x-lf.
(n.)y = x'{x-lf.
{m.)y = (x-mx-2r.
,. , x^+x+\
■ (IV.) y= —
, , x~-x + \
(vi.), = (--l)(--2)
(vii.) ?/ =
(viii.) y =
(^^■•)^= (x-4)
. . ^,? + \
[These curves are discussed algebraically and drawn to scale in Chrystal's
Jntroduction to Algebra, pp. 391-404. The student is recommended to com-
pare his results with those to be deduced from these figures.]
14. It c = — , -f ^„ prove that x — + y^= 2z.
a- b~ '■ ex •" cy
■X-
'' + x + -i
x-
' + X+1
{X-
-1){X-
2)
{X-
-1)(^'-
3)
{X-
-2){x-
4)
{X-
-l){x-
2)
lo. It i = tan~M -,,— -„ I, prove that X7^+ y ^r- = 0,
\x- + y-J ^ ox '' dy
52 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
16. The formula for the index of refraction // of a j^as at temperature 6"
and pressure ?. is ^_i=^._^_,
where /Xp^^the index of refraetion at O'^,
a^the coefficient of expansion of the gas.
Prove tliat the etl'ect of small variations 5^ and bp of the temperature and
pressure on the index of refraction is to cause it to varj^ by an amount
- _ /X|| - 1 / bp paSe \
"'^^ 760^ U + a^ (14 ai9)-/"
(l4a(?)-
17. U pv='\id is the ordinary gas equation, where d = l + al, writedown
the values of
. vv
... , dp
(n.).|'
(iii.) The approximate increase in the pressure due to a small decrease in
the volume, the temperature being unchanged,
(iv.) The ap[)roximate increase in the volume due to a small increase in
the tem])erature, the pressure remaining the same,
(v.) The approximate increase in the pressure due to a small increase in
both temperature and volume.
18. Assuming that the H.P. required to pro}icI a steamer of a given design
varies as the square of the length and the cube of the speed, prove that a '2%
increase in length, with a 7% increase in H.P., will result in a 1% increase
in speed.
19. The area of a triangle is calculated from measurements of two sides and
their included angle. Determine the error in the area arising from small
errors in these measurements.
20. Assuming that the area of an ellijise whose semiaxes are o and b inches
is wah sq. in., and that an elliptical metal plate is exj)anded by heat or
pressure, so that when tlie semiaxes are 4 and 6 inches, each is increasing at
the rate "1 in. per .second, prove that the area of the plate is increasing at the
rate of tt .s(|. in. pei' second.
CHAPTER VI
THE CONIC SECTIONS "-''
i^ 39. Ill this chapter we shall very briefly examine the pro-
perties of the Conic Sections, or the curves in Avhich a plane
cuts a Right Circular Cone. It is shown in the Geometry of
Conies that these curves are the loci of a point which moves in
a plane so that its distance from a fixed point is in a constant
ratio to its distance from a fixed straight line. The fixed point
S is called the focus ; the fixed line, the directrix ; and the
constant ratio, c, the eccentricity.
When e< 1, the curve is called an Ellipse ;
when e ^ 1, the curve is called a Parabola ;
when '>], the curve is called a Hyperbola ;
and the circle is a special case of the ellipse, the eccentricity
being zero, and the directrix at infinity.
I5 40. The Parabola (« = 1).
(i.) To Jiiid its equation.
Let the focus S be the point [a, 0), and the directrix the line
X + (( = 0 (Fig. 11).
Let P be the point (.'•, //).
Then since SP- = PM^
(,, _ af + ,/ = (,r + a)\
if- = ia.r.
* The stuilent is referred for a fuller discussion of the properties of the conic
sections to the books mentioned on p. 12.
5:j
54
THE CONIC SECTIONS
This is the equation of the parabola Avith the origin at the
point where the curve cuts the perpendicular from S on the
directrix. This point is called the vertex of the curve ; the
Fig. 11.
axis of X is called the axis of the curve ; and the ordinate L'SL
through the focus is called the Latus Rectum.
(ii.) The shape of the curve.
From the form of the equation of the curve we see that the
curve lies wholly to the right of the axis of //, and that it is
sjmmetrical with regard to the axis of o:
Also since
(If/ -la a
^ = — = / - when // > 0.
It follows that the tangent at the vertex coincides with the
axis of y, and that as we move along the curve in the direction
of o: increasing, the curve continually ascends, the slope getting
less and less the greater x becomes.
(iii.) The equations of the tangent and normal at (^x^, ?/q).
THE CONIC SECTIONS 55
fl il ^ ft
Since the value of — at (.'■,-,, ?/,.) is ~, the equation of the
iix 11 Q
tangent there is
y - % _ 2a^
X - .1'
■ 0
;'/o
or ;Vo(.'/ - //o) = 'M^' - ■'■o)>
which l:)ecomes ////q = 2a{x + x^), since i/^- = 4«3q.
Also the normal is the line
//o(.?; - ,ro) + -lad/ - >/q) = 0,
since this passes through (.>\^, ;//q) and is perpendicular to the
tangent.
EXAMPLES OX THE PARABOLA
1. Show that the curves x^— +4?/ are parabolas, and plot the curves.
'2. Show that the equation y = ax- + '2bx + c always represents a parabola,
and plot the curves
(i.) y = x^+'ix + B,
(ii.) 4Ly = x^ + Ax- 8,
(iii.) x = i/ + 7j.
Find also
(i.) The co-ordinates of their foci ;
(ii. ) The co-ordinates of their vertices ;
(iii.) The equations of their latera recta ;
(iv. ) The lengths of their latera recta ;
(v.) The equations of their axes ;
(vi.) The equations of the tangents at their vertices.
3. Find algebraically and graphically the minimum value of the expres-
sion X" - 2x - 4, and the maximum value of 5 + 4a; - 2x^.
4. The tangent at P meets the axis of the pai'abola of Fig. 11 in T, and the
normal meets the axis in G. Prove the following properties : —
(i.) AN = AT,
(ii.) SP =ST = SG,
(iii.) NG = 2AS,
and show that the tangents at the ends of a focal chord meet at right angles
on the directrix.
5. Prove that the line y = x+l touches the parabola y~ = ix, and that the
line y = mx-{ — touches the parabola y- = ia.r. Find the point of contact in
each case.
6. Find the equations of the tangent and normal at the point where the
line x=2 cuts the parabola x- = iy.
7. Find the equations of the tangents and normals at the extremities of
the latus rectum of the parabola y-—ia,r, and show that they form a square.
56
THE CONIC SECTIONS
8. Prove that the locus of the middle points of the chords of the parabola
y- = 4a,i-, wliich make an angle 6 with the axis of a', is the straight line
y = 2acot 0.
9. The chord PQ meets the axis of the parabola of Tig. 11 in 0. PM and
QN are the ordinates of P and Q. Prove tliat AM • A]Sr = AO^, by finding the
equation of the chord in its simplest form.
10. Tlie position of a moving point is given by the equations
.',' = V cos a . t,
y—v sin a. t- hgt'-.
Interpret the equations, and prove that the point moves on a parabola
whose axis is parallel to the axis of i/ ;
' X- sin a cos a c- sin- a '
9
whose vertex is at the point
whose directrix is the line 7/ = —- ;
and whose latus rectum is of length
S 41. The Ellipse (^<1).
(i.) To find its rijiiatidii.
2^
2v^ cos- a
9
Kin. V2.
Let the axis of ,'• be the axis of the ellipse {i.e. the line through
the focus perpendicular to the directrix) ;
and S the point (d, 0) ;
the axis of y the directrix.
Let 1' (x, if) be any point upon the curve.
Then SP2 = e^VMK
(.r-iJy + !/'=^e^A
THE CONIC SECTIONS 57
.-. ,f'-^(l - (T) - 2xd + If' ^ -d\
d Y 'I' <-l' '^^ 'i'c^
'■ - -- J + ■
\-<^' l-e" (1 - r'f 1-^2 (1 - e'r
Now change the origin to the point ( — ^—^, 0 j keeping the
axes parallel to their original directions.
The equation of the ellipse then becomes
[j- d^e-
x' +
(l-e2) (1-.^)^
I.e.
9 O
il-e- ti-c-
a^ =
dh^
and b- = 5,
1 - e^
Putting
we have
■., + 'f^= 1, where //- = ((-^(l - e^).
In this form the origin C is called the centre of the curve,
since it bisects every chord which passes through it. This is
clear, since if (x^, 1/^) lies on ^ + — = 1, so does ( - x^, - //^).
d de^
Also we notice that CS = ^ ; - d = , - ae,
i - e- \ - e-
and t^at CX = ;; :, = •
\ - e- e
From the symmetry of the equation
■'■- '/- ,
tt- 0-
it is clear that there is another focus, namely, the point {ae, 0) ;
and another directrix, the line .*; = -, with reg-ard to the axes
through the point C.
58 THE CONIC SECTIONS
The axis of x is in this case called the major axis, and the
axis of y the minor axis. The one is of length 2a ; the other of
length 1h. If h had l)een greater than a, the foci would have
lain upon the axis of //, and this axis would have been the major
axis. When a and // are given the eccentricity e is given by
//2 = „^(1 -,<^). {a>h.)
In the circle a - b, and e = 0.
(ii.) The shape of the nirre.
Since the equation involves only the terms ^'- and //-, the
curve is symmetrical about both the axes of ;<■ and //.
Also, since //- = l)-{ 1 — 5 j, we see that x must lie between
- a and + a, and that as ,r passes from - a to + a the positive
value of y gradually increases from zero to h, and then diminishes
again to zero.
The curve is thus a closed curve, lying altogether within the
rectangle x= ± a, y = ± h.
This is also evident from the property of Ex. 3, p. 59, where it
is stated that the curve may be drawn by fixing the two ends of
a string of length 2a to the points S and S', and holding the
string tight by the point P of the tracing pencil.
(iii.) The equations of the tangent and normal at (./q, y^,).
Since -^ + Vi = 1
a^ 0-
^ ■' ^1 = 0
a- b'' dx
Therefore the equation of the tangent at (.>„, y^^ is
?/^ % ^ _ ^%
X - .r^, ((%
which becomes
(^■-■'o),e+(//-//o)^^o,
2 ^,2
or
^-^ = 1, since ^ + §>,= 1.
a- b- a- 0-
THE CONIC SECTIONS 59
It follows that the equation of the normal is
or ^-^l^ = ryl'^^
«% h-ij ^
or — = «- - 0-.
^'o Ho
EXAMPLES ON THE ELLIPSE
1. Trace the ellipses (i.) 3a;2 + 4i/=12 ;
(ii.) 3(x- 1)2 + 4(2/- ■2y-' = 12;
(iii. ) x^+iy^=Sjj ;
(iv.) 4y-^ + 3y^=12;
and find the co-ordinates of the foci and of the extremities of the axes, the
length of the latiis rectum, and the eccentricity of each.
2. In the ellipse — + |s=l, show that the co-ordinates of any point may
be expressed as ,v = a cos d,y = b sin d ; and interpret the result geometrically.
3. P is the point {.'\, y-^) on the ellipse ^+r^ = l. Prove that SP = « + f.ri,
and S'F = a-eXi, and deduce that the curve is the locus of the point which
moves so that the sum of its distances from two fixed points is constant.
4. The tangent at P meets the major axis in T, and PN is the ordinate of
P, prove that CN . CT = CA2.
5. The normal at P meets the major axis in G. Prove that SG :SP = e,
and deduce that PG bisects the angle SPS'.
6. Prove that the middle point of the chord y = x + l lies u[)on //= — 5 a',
and that the middle points of chords jiarallel to y = mx lie upon the chord
, ''" .
?/ — m X, where mm + - ., = 0.
a-
7. If CP bisects chords parallel to CD, prove that CD bisects chords parallel
to CP (CP and CD ai'e then said to be conjiigaie diameter!^) ; and prove that
the tangents at P and D form with CP and CD a parallelogram.
8. If P is the point {a cos 0, h sin 6). prove that CD is the line «sin d.y
-f 6 cos e . X = 0, and deduce that CP- + CD- = a- + V^.
§ 42. The Hyperbola (ol). (i.) To find its equation.
Proceeding as in § 41 (i.) we obtain the equation
9 o
f/2g2
where we have written a- for ,- r-,?
{'" - 1)-
60
;ui(l
THE CONIC SECTIONS
h- for ., , i.e. for d-^n- - 1),
and d is the distance from the focus S to the directrix.
Fi<!. 13.
a
It follows that CS = nc, CX = , and that there are two foci
(',
and two directrices.
The line joining the foci S, S' is called the transverse axis of
the hyperbola.
(ii.) The shape of the curve.
The form of the equation shows that the curve is symmetri-
cal about both axes, and since //- = //-'( — , - 1 ) it is clear that .
cannot lie Ijetween - ((■ and + a, while since ,'■'-* = <i'{ 1 + p ), // can
have any value Avhatsoever.
If we write the ecpiations as
X- a'- X-
we see that, when .'• is numerically very great, ' :, is less than, but
THE CONIC SECTIONS (n
/ -
very nearly equal to .; ; and that for all points on the curve
'f . 1 t, ^'
-, IS less than — , ■
.(•"- fl-
Also the value of // decreases as .'■ passes from - x to - a,
where it vanishes, and it increases without limit from the value
zero at x = a as .'• passes along the positive axis of ,'•.
The shape of the ciu've is thus as in the figui'e. The lines
// = ± - ./■ are called the asymptotes, and the curve lies whollv
a
between those lines; while, as the numerical \'alue of x gets greater
and greater, it approaches more and more nearly to these lines
without ever actually reaching them.
(iii.) The eqvdtioiis of the tanrjent and normal at (,/;„ y^^ are easily
shown to be
and |(.' - .;,) + 'p - „,) = 0.
(iv.) The product of the perpendiculars from any point on the
curve to the asymptote's is constant.
The asymptotes are the lines // = ± - .'■. Then if PM, PX
are the perpendiculars to these lines from the point (.r^, //^j,
/; b
PM = — -r——, PN = — ,-—- •
\ 1 + -, \^ 1 + ~>
Therefore PM . PN = ^^f "J^ = 4^-,^
since ^ -'^- 1 .
a- ir
Hence PM . PN = constant.
Now when //- = a"', the asymptotes are at right angles, and
the eccentricity is \/2. In this case, by taking the asymptotes
62 THE COXIC SECTIONS
as axes, the equation x^ - if- = a~ is transformed to
2:nj = a-.
This equation is of the form ./// = ';-, a relation which is of the
greatest importance in Physics. We could obtain an equation
of the same form for any hyperbola referred to its asymptotes
as oblique axes.
EXAMPLES ON THE HYPERBOLA
1. Trace the hyperbolas :
(i.) •ix^-iy-=l2,
(ii.) 3(a;- 1)2-4(2/ -2)2 =12,
(iii. ) a;2- 4i/2=8i/,
(iv.) 4x'- 3i/=12;
and tincl tlie co-ordinates of the foci and of tlie points where each curve cuts
its transverse axis, tlie length of the latus rectum, and the eccentricity
of each.
2. Trace the rectangular hyperbolas :
(i.) rij=±4,
(ii.) u='i-±l,
and find the co-ordinates of the foci and of the points where the transverse
axis meets each curve.
3. Prove that the tangent at (x^, )/„) to the hyperbola .17/ = c^ is
a^2/o + 2/-^o = 2c2, and that the point of contact bisects the part of the tangent
cut off by the asj^mptotes.
2 ^
4. Pis the point (.'„ //j) on the hyperbola whose equation is —,-^=1.
Prove that ^? = cx-^-a, and 8'V = ex-^ + a, and deduce that the curve is the
locus of a ]ioint which moves so that the difference of its distances from two
fixed points is constant.
5. The tangent at P on the hyperbola ^-fr, = l meets the transverse axis
in T, and FN is the ordinate of P. Prove that CN . CT = a2.
6. The normal at P meets the major axis in G ; show that SG = eSP, and
deduce that PG bisects the angle SPS'.
o 'J
7. Prove that, in the hyperbola '-2-10 = 1, the middle point of the chord
y = x+\ lies ujion the line y = —,x, and that the locus of the middle points of
chords parallel to y — mx is the line y = in'x, where vivi' — —.■
a-
OF
THE CONIC SECTIONS 63
o ■>
X" if
8.' If OP and CD are two coiijiicrate diameters of the liyperbola —-'— = 1
•' '■ a- b^
[i.e. if each bisects chords parallel to the other), prove that if P lies upon
this curve, CD does not meet the curve, and that if D is the point wliere CD
meets the hyperbola -o-Vo^ - 1,
CHAPTEE YIl
THE INTEGRAL CALCULUS — INTECHATION
§ 4:3. Ix considering the motion of a point along a straight line,
we saw that if
is the relation l^etween the distance and the time, the velocity v
is given by '■ = ^^=/'(0,
and, in general, that the problem of the DitFerential Calcnlus is,
given the law in obedience to which two related magnitudes
vary, to find the rate at which the one changes with I'egard to
the other. The problem of the Integral Calculus is the inverse
one : given the rate at which the magnitudes change with regard
to each other, to find the law connecting them. In other words,
in the Differential Calculus we determine the infinitesimal change
in the one magnitude which corresponds to an infinitesimal
change in the other, when we know wliat function the one is of
the other. In the Integral Calculus we determine Avhat function
the one is of the other when the corresponding infinitesimal
changes are known. We have thus to find the function of .'•,
denoted by //, which is such that
The A^ilue of // which satisfies this equation is written y/(.r)</,r
and is called the iiifnjral ('//{■'') witk rcj/ard fa ./'.
d
E.I/. {i.)/.rd.r = ~, since ^-^('0
ax \ •! /
THE INTEGRAL CALCULUS— INTEGRATION 65
(ii.) y*sec-,rf/.'' = tan a;, since (tan a;) = sec-.*'.
In each of these cases we might have added any constant to
the right-hand side, since the differential coefficient of the
constant is zero, and the complete result would be
Jxclx = '^^ + C
/&QC?xdx = tan x + C,
where C is called the constant of integration.
It is thus evident that the equations
and F(.) =//{.)d.r
represent the same thing, and that the fuller statement of the
second would be
F(,r) + C =/f{x)dx.
Owing to the presence of the arbitrary constant ff{x)dx is
called the Indefinite Integral of J\x).
The geometrical meaning of the constant of integration is
that there is a family of curves all having the same slope as a
given curve, or parallel to it ; thus the curves
// = FOr) + C
are all parallel, when C is given different constant values.
§ 44. Table of Standard Integrals.
From this point of view of integration, as the process of find-
ing the integral is called, the first recpiisite is a table of the more
important forms. This table is obtained from the corresponding
results in differentiation, and any result in integration can always
be verified by differentiation. Later we shall see that there are
certain general theorems on integration which correspond to the
general theorems of dift'erentiation. These will help us to
decide upon the most likely ways of finding an answer to the
question which the symbol of integration puts to us ; namely,
What is the function whose differential coefficient is the given
expression l To answer this question is in very many cases
6
66 THE INTEGRAL CALCULUS— INTEGRATION
impossible ; but practice soon makes it easy to recognise the
cases which can be treated with success.
The following is the table of Standard Forms : —
(i.) fxMx = -, smce — = .c" (h 4= - 1)
^ ' -^ % + 1 ax \n+\J
(ii.) j ^ = ^°g ^'' ^^"^® ^ (^°g '') = ^
(iii.) Je'^Hx
a
1
(iv.) faHx = .
^ ' -^ loga
(v.) ycos xdx = sin x
(vi.) ysin 2'&' = - cos x
(vii.) ytan xdx - log (sec x)
(viii. ) ycosec iC(?a; = log ( tan- j
(ix.) /sec"xdx = tan x
(x. ) /cosech:dx — - cot x
/ . V f dx . -1 X / -ly\ \
XI.) ===^ = sin -or -cos -) (a2>a;2) L ,.
^ ^ .' \/a2 - X" "■ V «/ ^ ^ I Radian
f dx 1 -1 .>■ 1 -1 .-c I Measure,
(xii.) -i x = -tan -or cot
./ a-^ + .^^ a a a a )
(xiii-) I =^, = log(.^•+ Va;-^ ± a^)
(Unless otherwise stated, the logarithms are supposed to be to
the base e.)
The student is recommended to draw up a corresponding table
for the cases ivhere mx + n takes the pilace of x in this list.
§ 45. Two General Theorems.
(i.) f{cu)dx^cfiuh;
(ii. ) f{u + v)dx =J'udx +J-vdx,
c being a constant, and u, v functions of .t.
To prove these theorems it is sufficient to show that the
differential coefficients of the two sides of the equations are the
same, since in that case the answers to the questions which the
sign of integration puts to us are the same for both sides of the
equation.
THE INTEGRAL CALCULUS— INTEGRATION 67
They may be proved directly as follows : —
(i.) Let
f{x) =Judx.
Then
|-/w = »'
••• i <*■»='«■
.•. cf{x)=Jcudx,
the integral,
r/udx =fcudx.
(ii.) Let
f{x) =fudx,
F(.r) =fvdx.
by the definition of
Then _i(;(,,:),Fw}=|,.;W.|,F(,..)
.•. /(.*■) + F(.'-) =/{u + v)dx, by the definition,
.•. /iidx +/vdx -Jill + v)dx.
Ex. 1 . J{o-'3? + 2&a: + c)dx ~ a/a?dx + 2bfxdx + cf\ d.r *
= x + \og{2x-l).
3. [ '^^ = []_(J: L_Vtet
Jx--a? J 2a\x-a x + a)
if dx 1 / dx
2a J X - a 2a J x + a
= ^ loe I '■ ] , where xya.
2a * \x + aj^
4- ycos ax cos hxdx — -j [cos (« + b)x + cos {a - b)x]dx
= - / cos (a + b)xdx + - / cos {a - b)xdx
■ sin {a + b)x H- — r, sin (a — b)x.
2{a + b) ^ ' ' ' 2{a-b)'
§ 46. Integration by Substitution.
To prove that /f{x)dx =/f{x) • jj-dt, where x = <^(t).
* Jl . dx is usually written a,sjdx.
t This is an important example. Cf. (xii.) p. 66.
68 THE INTEGKAL CALCULUS— INTEGRATION
This important result, which allows us to change an integral
with regard to x into an integral in terras of another variable, may
be deduced at once from the rule for difterentiating a function
of a function.
Let y =/jlx)dx, and x = <f)(t).
From the relation between x and f, // is a function of t.
dy dy dx
dt dx dt
dji dx . dy ,
•'• irf^^'^dt^ «"^ce^-^.=/0.)
11 = f fix) — -df, bv the definition of an
integral.
The expressions under the sign of integration are supposed
given in terms of t.
This result may be written
(A.) Jf(x)dx ^ff{x) . '1^ . dt =/^[</>(/)]-|[<^(0]rf/.
The simple rule for " changing the variable " from .'• to / is :
Replace dx hi/ -^ . dt, and hi/ means of the equation connectinq x
dt ' ■ -L
and t, exp'ess f(x) as a function of t.
The advantaares of this method will be evident from the
following examples : —
Ex. (i.) J{a:i- + h)"dx. Vwt n.i- + h = u.
dx _1
du a
r r I if ?t"+'^ 1 ■"+^
and J(a.r + byd:r=Ju" . -. du=-\u"du = ~ ztt^-, ^Aax + b)
if. , cos n 1
Similarly (ii. ) y sin {ax + h)dx = -- I sin udu = = - - cos {ax + h),
f dx
(ill-) / . o ,o • Vut ax = ii.
-' \' a-x- - ir
. dx _\
dv a
J f dx /■ 1 ] , 1 j" du
and / , - = / / - • • du = - -,
J >y«V _ 1-2 J V?t^ - 63 a a j ^'„-i _ b-i
1 ,
= - log {ax + \'aV- - b").
THE INTEGEAL CALCULUS— INTEGRATION 69
(iv.) P^g'^-.,^,^, Vnt.r = c".
X
dx
clu
and / -? '^clx = / - • c« • dii = fudic = ^ u^
J X J e" J 2
= \\o^xf.
^^•^ \a-X^Y^ Put,.=co.s(^.
{\-x)i
dx
■ ■ d0= -''''''
f dx f 1 , . .^ ,,
j(l-a')Vl-j;'2 j (1 - cos ^) sin ^ ^
_ _ 1 f dd
•' sin-;5
.6
= cot-,
_ V ] - X'
l-x
(vi. ) Integrate the following exi^ressions : —
, x"
(a) x^'-^iax'^ + h); x\la\ + x^; ^^j-^.
^^) x^ + lx + 2 ' ..-^r2"J+ 2' P"""'S ^' + ^ = "•
(3) -7^^-r - ., ; z-^'^^, ^. putting .,_• + 2 =. M.
(^) / 9 „^ = ; /-' o ^ , - ' Pitting aa; + h = u. {ac)h-.)
s'ax^ + 2bx + c \'ax^ + 2o.r + c
1 COS CtJ
(f ) sin^ :<: cos^ a; : , — -. — : cot .'•, putting sin x = 7i.
iv) -2 5 ur-^^r ; -- o --o . putting tan a; = i(.
§ 47. Integration l)y Substitution — continued.
Although there are certain general principles that guide us
in the choice of a suitable substitution, the second form (B.),
p. 70, of the theorem of § 46 Avill often suggest what the
transformation should be. We have seen that
ffW)] ^^^W)¥t=/f(x)dx, where x = <f.{t),
70 THE INTEGRAL CALCULUS— INTEGRATION
and we may write this result in the form
(B.) //L>(.r)]^[</)(.v)]rf.; =ff{v) . du, where ^t = ^(x),*
as the particular symbol we employ is immaterial.
Thus in the case of the examples of last article Ave obtain our
results immediately — ■
e.g. (i.) /(ax + Hydx = - I (ax + i)" • -^ (ax + h)dx
= - u'hh, w
a J
here ax + h — u
a J
1 u»+^ 1
71 + 1 a
-, -. (ax + /0"+'-
a(%+ 1) ^ ^
(ii.) ysin- X cos xdx =y*sin- x -r;(sin x')dx
-/ifidti, where sin x = u
= - sin"* X.
o
(ill.) -. ■ dx = - . • T- • (1 + -'^0 • d-^
^ ^ ; 1 + x^ 5J I + x;' dx ^ ^
--] - , where m = 1 + x^
5.' u
= -\ogu
= l\og(l+o:^).
(iv.) jm^^^^iogM.
In this way it is easy to see that
f ax + h , 1 1 / o ^,
/ 9 ^j dx = - log (ax- + 2bx + c),
J ax^ + 2bx + c 2 *= ^ ^
* This can be verified ]>y starting with
//{vyiu,
ami putting u — <p(x), as in (A.)
THE INTEGRAL CALCULUS— INTEGRATION 71
since the integral may be written as
1 fdU , o ^7
i.e. - 1 — , where w = ax- + 2bx + c.
2.1 u
Also / — 2 — -. = I ;; ^, T^ • — (aa + h) • dx.
J ax^ + 2ox + c .' {ax + b)- + ac -¥ ax '
= I -s rz^ ■ du, where ii — ax + h,
J u^ + ac - b^
and this is one of the standard forms.
It follows that amj expression of the form
Ix + m
ax" + 2hx + c
may he easily integrated, since we can rewrite the numerator as
P(aa; + 6) + Q,
, ^ I ,^ am - lb
where P = - ; Q = .
a a
If higher powers of x occur in the numerator, we must first of
all divide out by the denominator till we ol>tain a remainder of
the first degree or a constant.*
The expression . may he reduced in a similar
fJax? + 2hx + c
way.
Ex. Integrate the following expressions —
1 1 1 x+1 2x + Z
(i.)
(ii.)
x^±4: ' a?x^±b'^ ' 4.r2 + 4x- + 3 ' 4a;2 + 4a;±3 ' 3 + 4x-a:2 '
X^ _ X'-X + l _ x-\ _ x^ + x + \
a;2 + 1 ' .x-2 + a; + 1 ' x- - 5x + 6 ' (.r -!)(./■- 2)"
11 1 a;+l 2x + 3
v/a;2 + 4 ' sld'x"±lr ' \/4a;^ + 4« + 3 ' v'4.>j- + 4a; ± 3 ' x/5 + 4.T-a;2'
* When the factors of the denominator are real, the method of Partial Fractions
should be employed.
72 THE INTEGEAL CALCULUS— INTEGRATION
§ 48. Integration by Parts.
The second important method in integration is called
integration by parts, and can he used only when the function
to be integrated is the product of two functions, one of which
can be expressed as a differential coefficient. This method
follows at once from the rule for the differentiation of a product.
^. (/ . , dv da
omce -r (uv) = u ^ + v ^,
ax dx dx
uv - \(u + V j-jdx, by the definition of integration,
r dv , r du , , . , _
r dv . f du ,
JU-— dx = uo -J r -- -dx.
dx dx
This result Avill be of use only if fv - dx can be more easily
evaluated than fu —- dx.
dx
For example —
(i. ) fx ■ log X ■ dx = ;^ I log « ^ (a;2) • di
dx
= ~{x' log X -Jx^ ~ (log x)d.
= - ix- log X -fxdx)
= -(^o:'\ogx-'^)
(2 1ogr.-l).
_x-
(ii.) y.'r • cos X • dx =Jxr % (sin x)dx
= x'^ sin 9:; - /"sin .'• ■ (.*•■-') • dx
= x^ sin X - 2ysin x ■ x • dx
= J? sin X + 2fx • -7-(cos x)dx
= X- sin ;*; + 2
/; cos
X -fcos, X -^ (x) • dx)
THE INTEGEAL CALCULUS— INTEGKATION 73
= X- sin X + 2{x cos x -J'cos xdx)
= .)? sin X + 2;/; cos x - 2 sin x.
In both of these examples this artifice allows us gradually
to reduce the integral to one of a simpler form, and in such
cases where powers of o: are associated with a trigonometrical,
exponential, or logarithmic term, it is of great value.*
An important expression which can be integrated by this
method is \';';- + a-.
We have
f ^'^V^' ■ dx = j \/:r^Va^ • J (./•) dx
r rl
= X v^,T- + a- - I X — 'Jx- + d^ • dx.
J dx
■ ■ - I \^x- + a- ■ dx = .(-■ six- + a- - \ -/ „ „ • dx
J J v.T^ + fr
dx
■ /■./■■^ + a^- a-
X \'x? + ft- - I - , ^ q
i \- a;-^ + ft^
/" ^ c dx
= ,/: V .r- + a?- - I s/ x? ■\- c? dx + ft- I / ., o'
; ; \'x^ + ft-
. •. 21 sJo? + ft"^ dx - X \/,i'- + ft"^ + ft- log (.}■ + V.;;- + d^)
I V a;- + ft- ft.v = -_ 4- — log (.>■ + \'x" + cr).
Ex. Integrate the following expressions : —
a'- log X ; a;^ e*^ ; .<; tan~^ x ; x- sin aa; ; Va'-^ - a" ; v'.-c^ - «^.
EXAMPLES ON CHAPTER VII
1. Integrate the following expressions —
N/aa; + & \/a; « + 3
,.. , 1 •-''■ - 1 a^ X*
(11.)
(iii.)
x{\-x)'' x-~-6x + -z' a;- + a; + l' x^-x + \'
1 2a; -1 x + 1
'Jx(\^x)' 'Jx^-Zx + 2' VaT^Tai + l"
Cf. p. 74 ; Exs. 11, 12, 13, 14, and 15.
74 THE INTEGRAL CALCULUS— INTEGRATION
2. Integrate the following expressions by parts —
sin"^./-, d? tan~^a', a;^ sin 4.'', o-} cos 3:/;, a:'" log x, x-c~^.
3. Prove that
1 1 a;-2
ar*+l 3(.r+l) 3(.r2-a- + l)'
and hence integrate the expression.
4. Prove that
1 111
+
(.j- + l)(a;-l)2 2(a;-l)- 4(,''-l) 4(a;+l)'
and hence integrate the expression.
5. Prove that
x-\ 2 1
(a;-2)(.Y-8) :>-^ «-2'
and hence integrate the expression.
6. Integrate the expressions x V 1 + x and — , by putting ,i- + 1 = v.-.
7. Prove that
dx 1^1 x\^'l
r = — r= tan~'
/,-
(1 + a;-) N^l - x^ \'2 """ s'l - x-
(pnt a; = sin 0).
8. Integrate the following trigonometrical expressions-
sin X
sin e' sin {6 + a)' sin ^ + cos d' cos"'* ^ s,/a^ tan- 9 + 6'-^' cos"^ a;(4 tan^^a:; + 3)
9. Show that, when a^b^,
f dx 2 _j/ /ci-h ^ x\
J a + b cos X- ^/„2— 6^ *^" ^ V a + ^, ^an - j.
Put a + b cos a; into the form {c + h) cos" '| + (« - Z*) sin- 1 •
Also integrate the expressions
1 1
5 ± 4 cos a; 4 + 5 cos a; 3 ± 2 sin .^; 2 + 3 sin x
10. Prove, by integration by parts, that
i; \ Car 17 & sin 5a; + « cos &a: „^
(1. ) /c"^ cos bxdx = 5 — rs e ,
r ■ \ r OT ■ I 7 a sin 6a; - & cos bx
(11. ) /c"" sin bxdx = 5 — ,„ ef".
11. Prove, by integration by parts, that
/■ • « ojo cos ^ sin"-^ ^ n -if. „ ^ ,
I am" Odd = + /sin"--6>(i?
•^ n n J
and hence show that
f ■ i n,a sin^^cos^ 3 . , n 3^
7 sill-' Bi/d = sin e cos ^ + s ^•
4 o o
THE INTEGRAL CALCULUS— INTEGRATION 75
12. Prove, by integration by parts, that
r r, nja sin 6 cos"-i d {n -I) f „ , ^ ,^
/ cos" ed0 = + ■ cos"-2 ddd,
-' 71 11 J
and thus obtain the value of/ cos'' dd9 andy'cos-' ddd.
13. Prove that
Jai^'c^dx - ,7;"e=^ - nfx''-'^e''dx,
and explain how this result may be used in evaluating such integrals as
fu^e^dx, Jx?e~'^dx, etc.
14. Prove that
y^n-i(iog xY^dx —Jif^c^ydij,
where x=ey, and explain how this result may be used in evaluating integrals
such as
fr'^{\og xfdx, J'x~\\og xfdx.
15. Prove that
x^ sin mxdx = - '— cos mx -\ — / a;""^ cos mxdx
m mj
a;" n ,. , . n.7i-l f „ <, . ,
= cos 7nx -\ — s a-""' sin mx 5 — I ^ sm mxdx,
m m- m'' J
and show how this may be used in evaluating such integrals.* Obtain a corre-
sponding result in the case of
fx" cos mxdx.
* Examples 3, 4, 5 are cases of the use of the method of Partial Fractions
in the integration of algebraic functions ; 11-15, of the method of Successive
Reduction. Cf. Lamb's Infinitesimal Calculus, §§ 80, SI.
CHAPTEE YIII
THE DEFINITE INTEGRAL AND ITS APPLICATIONS
i^ 49. In the last chapter we have considered the process of
integration as the means of answering the question : What is
the function whose difierential coefficient is a given function ?
There is another and a more important way of regarding the
subject, in which integration appears as an operation of sum-
mation, or of finding the limit of the sum of a number of terms,
Avhen these terms increase in number and diminish indefinitely
ill size. We shall examine integration from this standpoint in
the following sections.
§50. Areas of Curves. The Definite Integral as an Expression
for the Area.
Let //=,/(.'•) be the equation of an ordinary continuous curve,
and let us consider the
area enclosed between
the ordinates at P^C'^'o) ^o)'
and P(.r, y), the axis of
X and the curve Avhere
PqP is above that axis.
This area is obviously a
function of .r, since to
every position of P cor-
responds a value of the
area.
Let A stand for the
area PoMqIMP ; A + SA
for the area P^M^NQ ;
and let Q be the point (.'• + h\ // + %). Then if the slope is
76
THE DEFINITE INTEGRAL AND ITS APPLICATIONS 77
l)Ositive from P to the neighbouring point Q, we see by con-
sidering the inner and outer rectangles at P and the element
of area there, that
v/5.r<SA<(y + 8i/) 8.r,
and if the slope is negative the signs are reversed.
Hence in each case, when we let 8x approach its limit zero,
we have
thus A =/f{'i)dx + const. = F(.';) + C, sa}^
Also, since A vanishes when x = x^^, C = - F(.)'(,) ;
.-. A = F(,r) - F(.g.
This expression F(.'') - F(,*'p)
is an important one, and the symbol
J Xn
is used to denote it.
/ f(x)dx is called the definite integral of f{x) with regard to x
J Xq
between the liviits x^ and x, and its value is obtained by siibtracting the
value of the indefinite integral— ff{x)dx— for x ^ x^from that for x = x.
AVith this notation the area of the curve y =f(x) included
between the ordinates at (iq, y^) and {x^, //j), the axis of .'• and the
curve is equal to I f{x)dx, and it is clear that if the curve cuts
J .'ij
the axis between the limits x^ and ,r^, the definite integral gives
the algebraical sum of the areas, those above being taken positive,
those below the axis negative.
Ex. 1. To find the area of the part of the circle x~+i/ = a'-^ cut off Ijy the
lines ,T = 0, and x = Xi.
The required area = 2 / Ja'^ - x- dx.
J ^n
Now it is easy to show that
j s'^i^' djx^'^'^'^ + \ sin-i (■^) (cf. § 48) :
the area = x \'a^ - .'/- + a- sin-^ ( - j ,
78 THE DEFINITE IXTEGEAL
where we iise these square brackets to denote tliat we subtract the value for
x = Xq from that for ci: = xi.
If we take X(j = 0 and Xi = a we find the area of the semicircle as 7 ctr.
■1. To find the area of the part of the parabola y-=\ax cut off by the
lines x — Xq and x=x-^.
Here the required area = 2 / \/4aa- dx,
J ^0
/- /^^ n
= 4 N ft / \X dx,
9
and it follows that the area cut off by the latus rectum is ^ of the rectangle
o
upon LL' as base, with AS for its altitude.
3. Prove that the area of the ellipse -2 + |^ = l is wab.
«2 ' 62
4. Prove the following
O '
(1.) / =log(\/2 + l)= / -^—
J o^osx ° ' ./ 77 sm ;
4
W IT
(ii.) / sin-x.dx = j= I cos-x dx.
Jo 4 J „
... , /"'J dx _jr^_ j'2 dx
J a a"^ sivP'X + b"^ cos^^c ~ 'lab ~ J q a- cos'-u; + 6^ sin'^
(iv. ) / sin-^a; fte = '^ecosedd = ^~l
(V.) f'P--''^'
sm-a;
/dx _ TT
.5. Prove that when ?n and n are positive integers
TT TT
(iii. ) / '"' sin'" d cos" ^f/^ = '^^—— i ^ sin'"-'- B cos" ^rf^.
^0 ''(■ + "-'0
AND ITS APPLICATIONS 79
(iv.) I" sin^ecos^ede = — ^
{y■)
/ ^ sin« ^ cos« ^f^^ = ^^' ^' I", / " cos* edd = ^, ■
14. 12. 10./ 0 '-^l'^
In cases where integration is not possible there are various
approximate methods of finding the area. The expressions for
the area of a trapezium or a portion of a paral)ola give the
trapezoidal and parabolic rules, and we shall see more fully in
§§ 51-52 how the inner and outer rectangles may be applied.
The value of a definite integral may also be obtained by
mechanical means by the use of difterent instruments, of which
the planimeters are perhaps the best known.
Ex. Evaluate the following integrals by the trapezoidal method, i.e. find
the sum of the inscribed trapeziums instead of the inner or outer rectangles
as above : —
,12
(i. ) I .i'hlx, dividing the interval into 11 ecpial parts, and compare with
- 1
tlie result of integration.
Answers, 577i ; 575f .
.32°
(ii. ) I cos d.r, by dividing the interval into 6 equal parts, and compare
■^ 31°
as above.
Answers, -0148 ; "0149.
§ 51. The Definite Integral as the Limit of a Sum.
We have in the last article shown that the symbol I f(j:)dx
J ,-'0
represents the area between the curve y=f{x), the axis of x, and
the bounding ordinates. We shall now obtain an expression for
this area as the limit of a sum, and thus see in Avhat Avay the
process of integration may be viewed as a summation.
Let PqPi be any portion of the curve on which the slope
remains positive.
Divide the interval M^M^ into n equal parts S.t, so that
erect the ordinates m-^p-^, ni^'p.^, etc., and construct inner and
outer rectangles as in Fig. 15.
Then the difference of the sum of these outer rectangles and
80
THE DEFINITE INTEGRAL
the sum of the inner rectangles is {//^ - ^q)o'', and this may be
made as small as we
please by increasing
the number of inter-
vals and decreasing
their size.
Also the area of the
curve lies between
these two sums, and
tlierefore this area is
the limit of either sum
ss 8x approaches zero.
Now the sum of
the inner set of rect-
angles
M(^m, mj rrij
Fin. 15.
= 2 f(x^ + r8x)8x.
-f./(.r, + w-1.8,r).fe]
But the area is [F(;r^) - F{.'\^^)] Avhere F(,/') =/f{x)dx, and we agreed
/■'I
to denote this by / /(.')'/■'■.
... r f{.^)dx = Lts,=o X '^" ./(.'o + rSx) . &r,
.' ■,■() nSx = Xi-Xof )■ = ()
= lJs^^Q'Sf(x)8x, written shortly.
It is easy to remove the restriction placed upon /(.') that the
slope of the curve should be positive from P^ to P^ ; and to show
that this result holds for any ordinary continuous curve whether
it ascends or descends, and is above or below the axis in the
interval .'v to x^.
It is only necessary to point out that in the case of such a
portion of the curve y=f(x) as is given in Fig. 16, the area of the
portion of the curve marked 11 will appear as a negative area,
andif//(r>/r = F(4
' f{.r)d.r, or [Vih) - F{a)l
f
AND ITS APPLICATIONS
81
is equal to
(I) - (II) + (III).
Fig. 10.
Tlie importance of this result lies in the fact that many geo-
metrical and physical quantities {e.g. volumes and surfaces of
solids, centres of gravity and pressure, total pressure, radius of
gyration, etc.) may be expressed in terms of the limits of certain
sums. The problem of obtaining these quantities is thus reduced
to a question of integration. The symbol of integrationy' really
stands for the large S of summation, and it was in the attempts
to calculate areas bounded by curves that the Infinitesimal
Calcuhis was discovered.
It is also possible to start with the definition of the symbol
f{x)dx
J Xo
as the limit of a sum, and then obtain its value in terms of the
indefinite integral.*
§ 52. The Evaluation of a Definite Integral from its Defini-
tion as the Limit of a Sum.
It is instructive to see how, by algebraical methods, the values
of certain definite integrals may be obtained direct from this
summation.
* Cf. Lariil)'s Calculus, §§ 90, 91.
7
82 THE DEFINITE INTEGRAL
For example, in the case of the parabola
y = A
we can obtain the area, or the Definite Integral, as follows : —
y, Sx{Xq + rSxf
r = 0
= S4;<-o' + (^0 + ^''-y' + (^0 + 25,>f + {X, + n-l. a,>f ]
- i,x\nx^^^ ^-n . {n-\) . x^h: + — r-^ y^'')")
using the results for
1 + 2 + 3 + . . . + {n - 1), and 1- + 2^ + . . . + (« - 1)2.
Therefore, since nhx = (x^ - x^),
'' 2 ' 8x(x, + r8^f = x^ix, - X,) + x,{x, - x^ (l - 1)
r = n-l
Lt 2 8x{x^ + rhf
fix = 0 ) ?- = 0
J-'^ ft = c»
fix = 0 )
/i6x= xi-.ro )
= x^\x^ - x^) + .ro(.r, - .ro)2 + ^(:r, - x^f
o J '•'1 ■*'0
x-dx = — ^ . " •
Ex. Prove in the same way that
fim 1
/ COS mxdx = j^^ •
■ 0
AND ITS APPLICATIONS 83
§ 53. Properties of f{x)d:r.
J -('O
The following properties of the Definite Integral may be
deduced from either of the definitions of this symbol : —
I. Mdx = - f{x)dx.
J Xq J Xi
II. ( f{x)dx = { f{r)dx + ( /(..>/,r.
J 'J:o J .'0 -' f
III. The integral of an even function between the limits - a and
+ a = twice the integral of the function between 0 and a.
E.g. I xMx=2{ x}dx='^a^
ain-e . de = 2
sin-Ode = TT-
IV. The integral of an odd function between the limits - a and
+ a is zero.
E.g. 1 xHx = 0, I sin^dd = 0.
Similarly 1 sin"' 6 cos'^''+^ ede = 0,
m, n being positive integers.
V. In applying the method of " change of variable " to the
evaluation of definite integrals, we need not express the result
in terms of the original variable. We need only give the
new variable the values at its limits which correspond to the
change from x^ to x^ in the variable x, care being taken in the
case of a many-valued function that the values we thus alhjt are
those which correspond to the given change in x.
84 THE DEFINITE INTEGRAL
E.g. I ^W^^.dx
*2
a^ cos-OdO, putting x = a sin d,
Jo
2(2
2 V 2 ,,
§ 54. Application to Areas in Polar Co-ordinates.
AYhen the equation of the curve is given in polar co-ordinates,
the area of the sector bounded bj' 0 = 6^ and 0-9^ may be
sho's\ai to be
with the same notation as before. Hence if the curve is r =f(B),
the sectorial area is
If''
"J 9o
Polar co-ordinates offer the most convenient method of finding
the area of a looji of a curve.
For example, the lemniscate
r^ = a^ cos 20
has a loop between 0 - - ^ and 6 = -.
1 (■'
The area of this loop = - f-dd
AND ITS APPLICATIONS
85
rt^
COS 26(16.
J
4
the area of the loop = a^ cos 26cW (Cf. § 53, III)
a-
a-
sin2(?
Similarly, in the Folium of Descartes, whose equation is
a;3 + yZ =3 3axy,
there is a loop in the first quadrant ; and transferring to polar
co-ordinates we find that the area of the loop
1
7-m
n
JfSa cos 6 sin ^V^ /^
I ws¥T"sin¥/
0
n
9 2 r^ cos^^ sin^^
2'' (cos^^^^ + sin=*^)2 '^^
9 .
3 ...
f-
J 11
l + f
r, ■ df, putting tan 6 = t, (Cf. § 53, V.)
3 o
3a2
~ 2 ■
3
Ex. — Prove that the area of the cardioide r = «(l - cos 6) is n'^a^-
§ 55. Applications to Lengths of Curves.
The length of an arc PgP^ of the curve //-./(«) may be
regarded as the limit of the sum of the different chords into
86 THE DEFINITE INTEGRAL
which PgP^ is divided by the ordinates at m^, ???.,, . . . (cf. Fig.
15).
Hence
arc PqP, = L/5,=o '^\f(8x)^'T'(8i/Y
, dx= I' ^14- m'-dy,
J 3/0
since / 1 + rjl] will differ from / 1 + f^j " by a very small
quantity when &x is very small, and the sum of these differences
multiplied by 8x will vanish in the limit.
If polar co-ordinates are used, we obtain in the same way for
the curve r =f{0) the two expressions
since the chord is in this case \^'{8r)" + (rSd)-.
Owing to the presence of the radical sign under the sign of
integi^ation, the problem of finding the length of the curve has
been solved in only a limited number of cases.
Ex. 1. Prove that the length of the arc of the jjarabola y- = 4ax from the
vertex to the end of the latus rectum is equal to a[\^2 + log(\^2 + l)]
2. Prove that the length of the cardioide r = a{l - cos 6) is 8a.
§ 56. Volume of Solid, whose Cross-section is given.
If the section of a solid by planes perpendicular to the axis
of X is given and denoted by A, the volume of the portion of
this solid cut off by two such planes may be obtained by
integration, since this volume is readily seen to be "'
x=xn
or
* With the notation of § 49 we have
A8x<:dy<(A+dA)dx
and d\'
ax
AND ITS APPLICATIONS
87
As a special case, the volume of such a portion of the solid
formed by the revolution of the curve ij =f(x) about the axis of
o: IS
rr[f(x)fd.r,
or
- i/cb;
J J-O
and for revolution about the axis of //, we have in the same way
>n
77
oMy.
yo
Ex. 1. The portion of the parabola if=^ax from the vertex to the point
P(a;, y) revolves aljout Ox. Prove that the volume of the cup we thus obtain
is 2aTrx'.
2. Obtain the volume of a sphere by considering the rotation of the
semicircle x^ + y- = a^ aliout O.v.
3. Find the volume (i.) of aright circular cone and (ii.) of a cone in
which the base is any plane figure of area A, and the perpendicular from the
vertex upon the base is h.
4. Prove that the volume of a spherical cap of height h is irh\r - ■^), where
r is the radius of the sphere.
§ 57. Surface of Solid of Revolution.
It is easy to show that the surface of a right circular cone
whose vertical angle is 2 a and whose
generators are of length I is tt/- sin a,
and we can deduce from this that
the surface of the slice of a cone
obtained by revolving a line PQ
about Ox is equal to
27r . PQ . NR,
where NR is the ordinate from
the middle point of PQ.
Suppose then that an arc P^P^ of the curve y=f{^') rotates
about Ox, the area of the surface generated by P^P^ is the
limiting value of the sum of the areas of the surfaces generated
by the chords into which we suppose this arc divided. That is,
the area of the surface generated by PqPi
0
N
Fig. 17.
1 +
. 6x
di/\ -
do:
. dx,
where ?/=/(■'')•
88 THE DEFINITE INTEGRAL
n
This may be written 27r yds, by changing the variable from
J So
X to s, Avhere s is the length of the arc from a fixed point to the
point (./•, y).
When the axis of revolution is the axis of y, we obtain in
the same way the expression 27r I xds.
Ex. 1. Obtain the expression for the surface of a sphere of radius a.
Here we take the curve y= va- - x'^,
and the surface =47r| s'a^-x^ '\/^ + ~o'
■I 0 y
■,j dx
dx
0 r
a
iira I
0
2. Prove that the area of the portion of a sphere cut off by two parallel
planes is equal to tiie area which they cut off from tlie circumscribing cylinder
whose generators are perpendicular to these planes.
3. Prove that the area of the surface formed by rotating the circle of radius
a, whose centre is distant d from the axis of x, about that axis is Att-cuI.
§ 58. The Centre of Gravity of a Solid Body.
If a number of particles of masses m^, ?«.„ . . . are situated
at the points (x-^, ?/j, z^) . . . their C.G. is given by
_ 2(m^^) _ 2(r/? ,.//,) . _ S(m^g^)
and as we may suppose a continuous solid l)ody broken up into
small elements of mass 8711 whose centres are (x, y, z), we may
write these results for a solid body in the form
""= — M — ' y= — j.r— ' '= — M —
In many cases we can transform these expressions into
integrals which we can evaluate by the methods already
employed, though in general they involve integration with
regard to more than one variable, and these cannot be dis-
cussed here.
AND ITS APPLICATIONS
89
We add some illustrative examples : —
Ex. 1. The Centre of Gravity of a Semi-circular Plate.
Take the boundary of the plate along the axis of y, and suppose the
semicircle divided bj' a set of lines parallel
to that axis and very near one another. The
C.G. of each of these strips PQ' lies on the
axis of X, and therefore the C.G. of the semi-
circle lies on 0./;.
We thus have
X—-
2/ xydx
J 0
ira-
~2"
4 r
= — 9 I X Ja^ - X? . dx
-' 0
[-i(«-.^)'T
_4^
TTff
4a
OTT Fig. is.
and */ = 0.
2. The Centre of Gravity of a uniform Solid Hemisphere.
Let the axis of .'■ be the radius to the pole of the hemisphere, and suppose
the solid divided up into thin slices by a
set of planes perpendicular to this axis.
Then the C.G. of each of these slices
lies on this axis, and therefore the C.G.
of the hemisphere does so also.
Then
I xyHx
J ft
•■_•' 0
- 7r«'
3
. 3
x = -a
3
90
THE DEFINITE INTEGRAL
3. Prove that the C.G. of any cone or pyramid upon a plane base is one
fourth of the way up the line from the vertex to the C.G. of the base.
4. Prove that the C.G. of the upper portion of the ellipse —+% = l is at
a' ' 62-
■ib
the point ( 0, --
' OTT
§ 59. Moments of Inertia.
The moment of inertia, I, of a set of particles ?n^, m.-,, ....
with respect to an axis from which they are distant i\, ?•<,, etc., is
the expression
m-^r^" + m.^r.-,' + . . . .
and in the case of a continuous solid body we may express this as
I = Ltsm=o ^r-8m.
The radius of gyration k is defined by the equation
I = M/;2.
In many cases we may obtain the values of I and F by the
use of the methods of integration we have been discussing.
We add some illustrative examples.
Ex. 1. To find the radius of gyration of a thin rod of mass M and length
21, about an axis at right angles to the rod and passing through its centre.
Here
I = i/<5,„ _ 0 ^^^ • ^^'
= p I x-dx,
- -1
where 2Ip = 'M
= 2/3 / x'^dx
^ 2
_M.P
3 •
2. To find the moment of inertia of a solid circular cylinder about its
axis.
Here I = Lf^y^-Q 2r^5?n,
where din = ph {7r(?' + 8r)- - wr-\
= irph{2r8r+{dr)-}, where p is the vol. density.
A-
0
Fig. 20.
AND ITS APPLICATIONS 91
Therefore I = ivph I ?•- . 2
ra
= 2Tr ph I r
•^ it
'2rdr
(1
r^dr
0
But 7rpA«2 = M;
■ ■■ i=mJ'
3. Prove that the radius of gyration of a thin circular plate of radius a.
about a diameter as axis is - «-.
4
EXAMPLES ON CHAPTER VIII
1. Find the areas bounded by
(i. ) 2/ = sin 2a3, a; = 0, x= -.
(ii. ) y = e~^ sin 1x, a; = 0, .r = -.
(iii.) The hyperbola onj = a?, x = x\, x = x^.
(iv.) y = x^, a; = 0, :/; = 4.
(v.) 2/ = 2a-^, the axis of y, and the lines 7/ = 2 and y=^i.
2. Find the area of the part of the parabola ii — x--Zx + 2 ci;t off by the
X axis. What does / ydx here represent ?
•^ 0
3. Trace the parabola (?/ - aj - 3)- = a; + ?/, and find the area of the part of the
curve cut off by the lines x — 0 and cC = 4i.
4. Find the areas in polar co-ordinates of
(i.) The part of r = a^ included between ^ = 0 and d = 2Tr ;
(ii. ) A loop of each of the curves ?- = a sin Id, a sin 2>d, etc. ;
(iii.) A loop of each of the curves r = a cos 26, a cos 3^, etc. ;
(iv.) The part of the hyperbola r- sin ^ cos ^ = a'^ included between 9 = di
and 0 = 6.2 >
(v.) A sector of the ellipse -^ + ^ = 1 and of the hyperbola —2-fo — '^, the
centre being the pole.
(vi. ) Prove that the area between the two parabolas y'^ = ^ax and x^=iay is
16ft-
3
92 THE DEFINITE INTEGRAL
(vii.) Prove that the area between the two ellipses ^, + ^ = 1 and ^ + '^=1
is 4aJ tan~^-.
a
5. By substituting x -a cos, 6, 7j = b ain 9, show that the [lerimeter of the
ellipse of semiaxes a, b is given by 4a P ^/l - c- sin-^ . dd, and deduce that for
- 0
an ellipse of small eccentricity the perimeter is apjiroximately 2ira ( 1 - -
6. Find the lengths of the following curves : —
(i.) The equiangular spiral r = ae^ '^°* " from d = <) to d = 2w \
(ii.) The spiral of Archimedes r = ad from ^ = 0 to ^ = 27r ;
X X
(iii. ) The catenary y = '~^(e" + e "j from a; = 0 to a- = a ;
(iv. ) And show that the length of a complete undulation of the curve
II — b sin -
■^ a
is equal to the perimeter of an ellipse whose axes are 2n'«- + 6^ and la.
7. Find the volumes of the following solids : —
(i.) The solid formed by revolving the jjart of the line rc + i/=l cut off by
the axes, about the axis of x, and verify your result by finding tlie volume of
the cone in the usual way ;
(ii.) The spheroid formed by rotating the ellipse 9a:-+16?/'^ = 144: about tlie
axis of X ;
(iii.) The cup formed by the revolution of a quadrant of a circle about the
tangent at the end of one of its bounding radii ;
(iv.) The cup of height h formed by the revolution of the curve a^y — u?
about the axis of y ;
(v.) The ring formed by the revolution of the circle {x-a)~ + y-=b^ about
the axis of y ;
^2 y2 ;~2
(vi.) The ellipsoid -2+t2+-2==1.
And show that if Sg, Sj, S^ are the areas of three parallel sections of a sphere
at equal distances a, the volume included between Sq, S, and the spherical
boundary is - (S0 + 4S1 + S2).
8. The ellipse whose eccentricity is e rotates about its major axis. Prove
that the area of the surface of the prolate spheroid thus formed is
27r6(& + -sin-^c).
X _x
9. The catenary 2/ = - f (;" + g " j rotates about the axis of?/ ; prove that the
AND ITS APPLICATIONS 93
area of the surface of the cup formed by the part of the curve from a = 0 to
r
a; = a is 2Tra- 1
\ ^
10. The cardioide ?- = a(l - cos 0) revolves about the initial line ; prove that
32
the surface of the solid thus formed is -r- tto^.
0
11. Find the C.G. of the following :—
(i.) A thin straight rod of length I in which the density varies as the
distance from one end.
(ii.) An arc of a circle of radius a which subtends an angle 2a at the centre.
o o
(iii.) A quadrant of the ellipse —, + '-jT,= 1-
(iv.) A circular sector as in (ii.)
(v.) The segment of the sector of (iv.) bounded by the arc and its chord.
(vi. ) A thin hemispherical shell of radius a.
12. Find the moments of inertia of each of the following : —
(i.) A thin straight rod, about an axis through an end, perpendicular to
its length.
(ii. ) A fine circular wire of radius a, about a diameter,
(iii.) A circular disc of radius a, about an axis through its centre perpen-
dicular to the plane of the disc.
(iv.) A hallow circular cylinder of radii a, b and height h, about its axis,
(v.) A sphere of radius a, about a tangent line.
(vi.) (a) A rectangle whose sides are 2«and 2b, about an axis through its
centre in its plane perpendicular to the side 2a ;
(/3) about an axis through its centre perpendicular to its plane,
(vii.) An ellipse whose axes are 2a and 2b,
(a) about the major axis a ;
{j3) about the minor axis b ;
(7) about an axis perpendicular to its plane through the centre.
iV.5.— The case of the circle follows on putting a = b.
(viii.) An ellipsoid, semiaxes a, b, c, about the axis a.
JS\ B. — For the sphere a = b = c.
(ix.) A right solid whose sides are 2a, 2b, 2c, about an axis through its
centre perpendicular to the plane containing the sides b and c.
A^.^.— Routh's Rule for these last four important cases can be easily
remembered : —
/ sum of squares of perpendicular \
Moment of Inertia about an axis\ _ \ semiaxes /
' of symmetry ^ -mass 3, 4, or 5
The deno7ninator is to be 3, 4, or 5 according as the body is rectangular,
elliptical, or ellipsoidal.
Cf. Routh's Eigid Dyna.mics, vol. i. p. 6.
ANSWERS
CHAPTER I. (p. 12)
1. (i.) x^-\-f-='^^. (ii.) ..-■=!^^.
(iii. ) ar* + 2.«Y + y* + 2a'^{y'^ -x-) + a* - c^ = 0.
2. x' + 4j/-ll = 0.
13 19^
^- • 11' 11
4. The parallel lines through 0 are
3a'-2?/=:z0, ix + y^Q, 19;/;+13j/ = 0.
The perpendicular lines through 0 are
2a' + 3i/ = 0, a;-42/ = 0, 13^'-19?/ = 0.
The parallels through (2 . 2) are
3.«-2!/ = 2, 4a; + 2/ = 10, 19.i' + 13?/ = 64.
The perpendiculars through (2 . 2) are
2./' + 3?/=10, ;(;-47/ + 6 = 0, IS.-.-- 19y+ 12 = 0.
5. a; + 3//-7 = 0.
6. 7a; + 7?/ -36 = 0 is the bisector of the acute angle.
a; -y- 12 = 0 is the liisector of the obtuse angle.
7. (i.) (1.2), (3, 4), (5, 3).
Si.) I -3, I
(iii.) The internal bisectors are
x-y+1 _-x-\-iy-7 x-y + l_-x-2ij + \l x- 4y + 7_.>J + 2y- 11
\J2 s'l7 ' n'2 VF ' n'17 ^5
The external bisectors are
x-y+l _x-iy+7 x ~y+l _x-{-2y -11 x -iy + T _ - x - 2y + 11
n/2 ~ \'l7 ' \/2 ~ V5 ' v^r7 ~ Jl
8. If the points (0, 0), (2, 4), (-6, 8) be called A, B, C respectively, the
equation to
(i.) BO is a' + 2?/ -10 = 0,
to CA is 4a3+32/=0, *
to AB is 2x-y = 0.
(ii.) tan A = 2, tanB = oo, tan C = --
95
96 INFINITESIMAL CALCULUS
(iii.) Median through A is y + 8x~0,
Median through Bis ?/ - 4 = 0,
Median through C is 6.>' + 7y~ 20 = 0.
(iv.) The perpendicular from A on BC is the line AB ; its length is 2 ^'5.
The perpendicular from B on CA is the line 3x-iy+ll = 0; its
length is 4.
The perpendicular from G on AB is the line CB ; its length is
4 ^,'5.
(V.)
x + 2y = 0,
4a; + 3?/ -20 = 0,
2x-y + 20-0.
(vi.)
4
(vii.)
/- 2(3-^/5) 2(4+ v5)
^ 3 + V5 3+^5
(-3, 4), ( -1.4
CHAPTER II. (p. 23)
4. y-9x+\6 = 0.
6. u-gt; -(J.
7. 2Tn-h5r.
8. 5V = 47rr25r, 50-27, 502-66.
The proportional errors are 1 : 1607r : 16007r.
9. (a + 2U) 8l = {a + 2bt)SL
12. — 7= feet per second.
\^3
CHAPTER III. (p. 31)
dy^B{x-inx + l)
d
X
s
2?;^
... , dy a-x
f'.^' \/2ax-x^'
,... . dy 2(z + 3
(111-) 31.=
(^a- 2v'(a;+l) (a;+2)
3^=(x + a)i^-
dx
dy 1
(iv.) ^=(x + a)i^-i(a;+i)''-i(y/ + f/).;f + ^a+;ji.
(v.) , -
dx {l-x)\/\-x^
(vi.) ^~^yi.i{qa-pb + {p-q)x).
, .. . ^ , {b(n-m):rf" + na)
(vn.) .>'"-i — -.
{Ox'" + uj-
(viii.) 7/(?u"-i(H-Jc" )"'-!.
(ix.)
\/x*-a*
ANSWERS 97
(x. ) - x{(x- + a^) -i + (a;2 - a^) " ^}-
. . . 3.x,'-'
(^1-) Ti'
(Xll.) -3-
{l+x + x^)^{l-x + x^)^-
2/0
(ii.) --«.
2/0
(lU. +-:3— •
«"2/o
(iv.) -■^.
ii'O
4. 7 '96 miles per hour.
5. 8 miles per hour ; 4 miles per hour.
' dv V
9. When the pressure decreases, the volume increases, and conversely.
CHAPTER IV. (p. 38)
1. (i. ) 3 sin a; cos a;(sin a; - cos cc).
(ii. ) sec *x.
.... . 4 sin a;
(in.)
(iv.)
(V.)
^'^^•^ (l + cosxf
3. (i. ) x^-\m sin (a;") + «,<;" cos (a;»)].
(ii. ) x"^~\tn cos (a;") - nx'^ sin (a;")].
(iii. ) af^-\m tan (.«") + ?ia;" sec ^(.x")].
4. (i.) 2ajtan-^a'.
(ii.) sin ~'a'.
1
cos
^x'
-4
cos a;
sin '^x
•2.
cos a;
(1-
sin x)"^
2
sin a;
(iii.)
2^x{\+x)
^^^■' i + Zx^ + x''
2(1 +a,''^)
5. rtw sin w/, aus' cos oit.
6. x = 2aw cos" — ; y-=aw sin a;<.
Jc= - aoo" sin w^ ; ij — cuxj^ cos w^.
The direction of motion at time t makes an angle -^ with tlie axis of x.
8
98 INFINITESIMAL CALCULUS
CHAPTER V. (p. 50)
1. (i.) e'(l+,r). (ii.) x"'-h"^{m + nx). (iii.) {a + be + cax)e'^+<'.
(iv.) e''^'''~^^( sm-h'.-r—;^=^].
^ ' \ Vl-W
2. (i.)2a'ci+^. (ii.) 2.i'e«^"(l+aa;2). (iii.) «'"-^e''^''(m + Ma«").
(iv. ) a-™-^a^"('»i + ?ia;" log a).
3. (i.)x"-(l+mlog.). (ii.)^^^i~- (iii-) 2;^.
-6a; ^/x'^+l+a- . 1
^^^•^l-a=2)(4-a;2)- ^^-^ a;x/^;vr" ^"^''^l -a;) Vi"
, ... 4a;-3 .... a^ ..... 2-5a;
2V(2a;+l)(a;-2)' (a2 + a;2)f' ■V(a;-1)'''
(iv.) .7:^(1+ log a;).
(^•) ' ..»»+L^ (^1-) log—-- -^ 1+ ^
cos'^+^Tia; ^ '' \ ^ X x + 1
X
11. (i.) tan a.
(ii. ) tan n0.
(iii.) -cotnd.
(iv. ) cot n6.
(v.) -tan n9.
r-r- is the tangent of the angle between the radius vector to the point
(r, Q), and the tangent to the curve at that point.
13.
(i.) g=(3a--l)(a;-l) . Max. at Q, ^.
llin. at (1, 0).
(ii.) V^ = a;(5a;-2)(a;-l)2 . Max. at origin.
Min. at ("4, -03456).
(iii.) ^==2(a;-l)(a;-2)(2a;-3) . Min. at (1, 0) ; (2, 0).
Max. at (I ^
^^''•)£ = ^"|2-Max.at(-L -1).
Min. at (1, 3).
(v.) $ = 2t-£^-^^. Max. at(-l, 3).
Min. at ( 1
' 3y"
Min. at (1-4, --06) nearly.
ANSWERS 99
dx (.r-l)V-'-2)2
(vii.) J- -/,^"!'-,^|^fj2 • Min. at ( - -9, -16) nearly
Max. at (1-4, 18-2) nearly.
(""•) 7ir='jt-Wi^-i? ^"^ *"^'°^"S points.
IX.) -f-= , -TTij— . Max. at (l-o, '1) nearly.
dx {x -\y
'\l\\\. at (6-45, 9-9) nearly.
(X.) '^ = ^-'1. . Min. at (1-26, 1-89) nearly.
,., 4'.
<"■) I
.... , . R^-
(ill.) op = ^^ov.
(iv.) Sv^—8t.
P
, , ^ R(H-a<)^ aK^,
(v.) 5»= - 5^-i — -Iv^ — U.
EXAMPLES ON THE PARABOLA (p. 55)
(1)
(2)
(3)
2.
Foci
(— !>
(-2, -2),
H)
Vertices
(-2, -1*,
(-2, -3),
i-H
Latera recta
3
2/= -2,
x = 0.
Lengths of recta
1,
4,
1.
Axes
if = - 2,
x= -2,
1
Tangents at vertices
y=-h
2/= -3,
1
^■=-4'
.3.
-5. 7.
5.
(1,
2), (;
a
2fA
6.
x-y
-1 = 0,
x + y-3=^0.
( .
x-y
+ a = 0,
x + y-3a — 0.
x + y + a — Q,
x-y - 3« = 0.
8 a
100 INFINITESIMAL CALCULUS
EXAMPLES ON THE ELLIPSE (p. 59)
1. The foci, extremities of the axes, length of latus rectum, and eccen-
tiicity are for
(i.)[±1.0], [±2.0], [0+N% 3, I
(ii.)[2.2], [0.2], [3.2], [-1,2], [1,2+^/3], [1,2-^^3], 3, ~.
(ill.) [± ^/3, 1], [±2,1], [0.2], [0.0], ^, ^.
(iv.) [0±1], [.)±2], [±v'3.0], 3, 1
EXAMPLES ON THE HYPERBOLA (p. 62)
1. (i.) (±v'7, 0):(±2, 0) :3:^.
(ii.) (1 ± v'7, 2) : (3, 2):(-l, 2):3:^.
(iii.) (0, - 1 ± V5) : (0, 0) : (0, - 2) : S : ^'5.
(iv.) (±n'7, 0) :(±V3, 0):i|^: -^-|\
2. (i.) (±2^2, ±2\/2) : (±2, ±2).
(ii.) (±2^/2, +2x/2):( + 2, +2).
CHAPTER VIL (p. 73)
1. ,. ,(a;-a)* 2sJax'-^'b 2 ,-,„ ^
(1-) —i~ '-^-a = 3 V.^'(3 + a:) : «-log(a; + 3).
,.. , , X . {x~2f (x-lf 2 -i/2,7-+l\
(iii.) sin-i(2a;-l); 2 Vcc'^- 3.7J + 2 + 2 log (.r-^+ Va;—3a.- + 2) ;
V(a;2 + .'• + 1) + ^ log ('a; + ^ + \/x^ + .r + l \
2. a,-.sin-b + v^r^''; '^ts^n'^x-^. x^ + ^log{l+x') ■ cos4c>/m^') +
sm ix . -^ ;
o
/9,7;--2\ . „ 2x .^™+i .r'«+i
ANSWERS 101
3- ;t log , — ——2 + ^5 tau
'■ -2
5. log
2a;-l 4 ^ \.r-l,
(a; -3)2
a;-2
6. (i.)2(l+,)»{|-A}. (ii.)log-^|
Va; + 1 - 1
Vx + 1 + 1
8. log tan -. log tan ^ . —77: log tan ~ +
1 — 1 •, 2 sec X - 1
-log(atan^+ ^Han^^ + n 4l°g2sec^-+ l'
f dx 2._,fl.x
[^^^^ = 1 tan- fa tan:;
j5-4cosa;3 \ 2
/cfa' _ 1 ,
4 + 5cosa;~3 °^
j 4 - 5 cos a; 3 ^
,7'
3 + tan ^
3 - tan -
X
1 + 3 tan 2
1 - 3 tan -
2
f_^_= 2 tan-\/ltanf'|-'^y
j 3 + 2sina; ^5 V 5 \2 ^j
/" cgx _ 2 /^). ^\
J3-2sina;-V!*^''"'^^*^^i2~4J-
i dx ^2^^ v/5 + tan(|-^)
J 2 + 3 sin a; ,^5 ° ,- /a; 7r\
v5-tan\^2~ 4
i 2 - 3 sin a;
/- / a; TT
_^ ^l+v^5tan(^2"4
v/5 * ,- (x V
1 + v5 tan I 9 ~ T
10 1 2a • a , 2 . a cos 3^ sin ^ 3 „ • /I 3
12. 5 cos 2^ sin ^ + - sin ^ . ^ + t cos ^ sin ^ + k '
o 3 4 4 8
^" „;. ^^ _,_ " .,„-! .„„ „„^ «(?l^ Ln-2 ,
15. — sin mx -\ — r, a'""^ cos mx — ^ — ^ — - \ a;"~- cos mxdx.
m m~ m-
102 INFINITESIMAL CALCULUS
CHAPTER VIII. (p. 91)
1. (i.) 1.
(ii.) jlye •■^ + 1
(iii.) a- log -.
(iv.) 64.
(v.) 3f2§-J
2. -: the difference between the area bounded by the rc-axis, the //-axis
and the curve, and the area which lies on the negative side of the aaxis.
o 343
^- l2'-
4. (i.)^-^^
.... 7r«- TTrt'-^ 7ra-
("•^ ^; T^' 4,r
.... . TTtt" TTrt'^ ira?
o 1- 4'/6
,. . o 1 tan p-7
(iv. ) fr log , -".
^ ' '= tan 6-^
(y \ — tan-i ^^ ^^'^^ ^o^an^)
ah . ^^ {b + a tan O^) (b - a tan 9^)
4 ° {b + a tan ^J (6 - « tan ^2) '
,. /. ■! / 2 TT cot a , ,
0. (1.) a sec a (c -1).
(ii.) ^('(27rv'r+4^+log (27r + \/r+45f-)Y
(iii.)|(€-0.
(ii.) 487r.
..... 57r«* ir'^ci?
(iv.) '- irh^a^.
(v.) -laV-Tr-.
(vi.) - 7ra6c.
o
2
11. (i.) ,-.-: from that end.
•J
(ii.) On the radius bisecting the arc at a distance from the
centre.
ANSWERS 103
(ni.) x=—, y=-^-
TV OTT
,. ^ 2a sin a .
(iv.) On the radius bisecting the sector at a distance — trora
o a
the centre.
O SlU OL
(v.) On the bisector of the chord at a distance ^a -.
^ ' 3 a- sm a cos a
from the centre,
(vi.) The middle point of the radius perpendicular to the base.
12. (i.) I M~^ . (rod of length '2z).
(ii.) ~ Ma^-.
(iii.) ~Ma2.
(iv.) ^^{a^ + b%
(v.) ^Ma2.
(VI.) (a)-g-:(/3)M(^— g— j.
(vii.) (a)M|':(/3)M~:(7)M(^
(viii.)M(^^).
(i.)M(-±^).
4
THE END
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