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I. F. Quinby Horatio Nelson Robinson.

# A new treatise on the elements of the differential and integral calculus

. (page 19 of 22)

since cos. Xj when x is indefinitely increased, does not con-
verge towards any determinate limit.

For another example, take T â€” , in which a and b are

J _n X

any two positive quantities whatever. Since - becomes in-

X

finite for the value a; = 0, which is â€¢comprised between â€” a
and + 6, we put

/-f6 dx ,. (*-adx , ,. /â€¢H-6 dx
â€” â€” lim. / â€” + lim. / â€” ;
^a ^ J -a ^ J -^{3 OC

a and ^ being numbers numerically less than a and b respec-
tively, and the limits indicated being those answering to
a = 0, /3 = 0. But

The first term Zf- j in the value of this integral is deter-
minate; but, since the variables a and /3 are entirely inde-
pendent of each other, the term lim. l[-] does not converge

towards any fixed limit, and the integral is therefore indeter-
minate.

231. When the integral of Xdx is required, and X can be
developed into a converging series,

X=t^i+t^2 + 2^8+... + Mâ€ž + r^ (1),
we shall have, after multiplying by dx, and integrating between
the limits a and 6,

r Xdx= f Uidx+ f u^dx-] + r u^dx-^- f r^dx (2).

49

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S8G INTEGRAL CALCULUS.

If Bcrics (1) is converging for xziza^ a? = 6, and also for
all values of x between a and 6, we may assume r^ <^a;
a being less than any assigned quantity when n is taken suf-
ficiently great Whence

r r^cir < r adxj or f r,cir < a(6 â€” a).

Therefore j r^dx will decrease without limit when n is in-
creased without limit ; whence the series

r UidX"^ j u^dx-^ â€¢â€¢â€¢+/* Uf^dx
is converging, and its sum is the expression for f Xdx. The

J a

fixed limit h may be replaced by the variable x, provided no
values of x are admitted which fail to render series (1) con-
verging. We should thus have

/' Xdx=i \'uidx-\- ^'u^dx-^- â€¢â€¢â€¢ + f'u^dx (3).

232. Formula 3 of the last article still holds true for x:=by
even though the series t^j + ^2 + ^a + â€¢ * * y which is supposed
converging for x <^b, becomes diverging for x = b, if, at the
same time. Series 2 is converging.

For, however small the quantity a may be, we have

/Xdx:=: f Uidx-}- f ~ U2dx-\''" + f u^dx.

The two members of this equation are continuous functions
of X, and are constantly equal ; hence their limits for a =
must be equal, and therefore

/Xdx=i f Uidx+ f u^dx-^ â€¢â€¢â€¢ + r Uf^dx.

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INTEGRATION BY SERIES. 387

If the series

/(x)=/(0) + ^/'(0) + ^2/"(0) + ..., .

to which the development of f{x) by Maclaurin's Formula
gives rise, is converging, we shall have

Jf(x)dx = C+ 0/(0) + ^2 /'(O) + OT3/"(0) + â€¢ â€¢ â€¢ ;

and, if it is wished that this integral should begin with a? = 0,
G must be zero ; and we then have

jy(x)dx = x/(0) + ^/'(x) + ^/"(O) + . . .

Example 1. f ^ = Z(l + x).
By division, or by the Binomial Formula, we have

^ =l^X + X^â€”X^-{ zfca?Â»-^::F^'* "

,,^ , . x^ . x^ x^ ,x^ f^xx^dx

When X is numerically less than 1, positive or negative, the

series \ 'â€”X'\-x'^ â€” x^..,is converging, and therefore so also

X x^ x^
is the series x â€” ~ -[- - ... between the same limits for

^ Â«> 4:

x: hence, when x^^if

X* . X

lil + x) = x - ^- + ^ - ^

Â»ar X^dx

It may be shown by direct demonstration, that / â€”-pâ€”

converges towards as n approaches oo .

x^
For, if X is positive, we have iâ€” 7-â€” < a?** : therefore

JL "Y" X

Jo 1 +^ V n + 1

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388 INTEGRAL CALCULUS,

Now, as n increases, â€” -p^ approaches ; and consequently,

in stopping with any term of the series, the error will be less

than the following term, and will be additive or subtractive

according as the last term taken is of an even or odd order.

If X is negative, and a denotes a number greater than x,

cc^ x^

but less than 1, we have -z < q ; and therefore

' 1 â€”a; ^ 1 â€” a'

/:

X^dx . 23"+^

'o 1-0? ^ (n+l)(l-a)
The limit of the second member of this inequality for n = oo
is zero. In this case, the error is always numerically additive.
When oj = 1, the series 1 â€” a: + aj^ â€” - a;' -[- â€¢ â€¢ â€¢ is no longer

X X

converging ; but the series x r- + -q- â€” â€¢ â€¢ â€¢ is (Art. 231),

1 o

and will represent the value of Z2 : hence

1 a?*+^

We have -â€” j â€” ^ = 1 â€” a?^ + o;^ â€” a?* â€¢ â€¢ â€¢ liz a?** ~ ^ T :=â€” i â€” Â» ,
\-\-x^ ' l-|-a3^'

n being an odd number, and positive. Integrating, and taking

for tan.~^a; the least positive arc having x for the tangent, we

find

, x^ , x^ , x^ px x^'^^dx

3'5 n ^ J Q l+x^

The series 1 â€” x^-^-x* â€” x^... ceases to be converging for

flj = 1 ; but the series x __{ - -... is still converging for

O

this value of x : hence

X -1 ^1 1 I 1 1

tan. '^ X =: - = 1 â€¢â€¢â€¢

4 3^5 7 â€¢

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INTEGRATION BY SERIES. 389

Ex. 3. f -y== - = smr^x.
We have

From this, by multiplying by dx, and integrating, we find

. , , 1 a;' 1 3 a?* 1 3 5 ir^

Bin-a. = a:+^y + 2 4T + 2 4 6T-'

a converging series when x^^i') since Series 1 is converging

between these limits.

The series *"

-,1 â€ž,13 .,135 g

1 4- - x^ A X* A ~ x^

^2 ^2 4 ^2 4 6

is not converging when x=:l', but since, for a? = 1 = sin.

the series

^"^2 3 "^2 4 5 "^2 4 6 7
is converging (Art, 231),* we have

7r_ 1 1,13 11 3 5 1

2 "^2 3 "'"2 "4 5 "'"2 "4 6 7"^

A still more converging series is found by making

whence

1 . 7t.

7r_l 1 1 13 1
6 ""2 + 2 2^3 + 2 i 2^5 +

â€¢ Space does not allow the proof of converjrence or divergence when these con-
ditions are asserted relative to the series involved in the last three examples. (See
Art. 68.)

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390 INTEGRAL CALCULUS.

233. By integrating f{x)dx by parts, we have-
^f(x)dx = x/{x) - fxf{x)dx,

fx/'(x)dx = ^ fix) - J '^/"{x)dx,
fx/"ix)dx = '^/"ix) -J'^^-/"'{x)dx.

The combination of these results gives

^ 1.2. .71 '^ 1.2. .71 J ^ ^ '

This is the series of John Bemouilli, and may be advanta-
geously used in many cases : for ex.ample, \if{x) be a rational
algebraic function of {n â€” 1)*** degree, /^"^ (a;) is 0, and the
series will terminate ; or there may be cases when

Jx''f''\x)dx

can be more readily found thsui f/(x)dxy or when only an

approximate value of \f{x)dx may be required, and the in-
Â«/

tegral \ x^f^'^\x)dx may be small enough to be neglected
Â«/

without sensible error.

23d. Assuming Jf{x)dx = cp (x), and making a? = a; + A,
we have, by Taylor's Formula,

cf(x + h)-,f{x) = J>.f'{x)+^~<p"{x) + ... (1).

But, because J/(x)dx = tf (x), we have

/(x) = <r'{x), fix) = cf"{x), f"{x) = <f"'{x).

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INTEGRATION BY SERIES. 391

These values, substituted in (1), give

^{x + h)-^(^x) = hf{x)-\.!^f'{x) + ^f"{x) -{....

In this series, making x=zay h = b â€” a, and denoting by
A^j A^j A^..., what/(a;), /'(a;), /^'(a?)..., become under this
supposition, then qp(a; + A) â€” 9 {x) becomes

and we have

â–  jyi^x)dx = A,{b-a)-\.^^{b-af+^^{h-af...

This series enables us to find the approximate value of the

definite integral i f(x)dx when 6 â€” a is suflBciently small to

make the series converging. When this is not the case, or
when the series does not converge rapidly enough for our
purposes, put 6 â€” - a = na, and take the integral successively
between the limits a and a -|- a, a -}- a and a -(- 2a, and so on,
denoting the results by

then (Art. 224) we have

+ (i?, + C, + i),+ ..-)aÂ»+...,
a series that may be made to converge as rapidly as we please
by making a sufficiently snialL

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(

SECTION V.

GEOMETRICAL APPLICATIONS.

QUADRATURE OP PLANE CURVES REFERRED TO RECTILINEAR CO-
ORDINATES. â€” QUADRATURE OF PLANE CURYES REFERRED TO
POLAR CO-ORDINATES.

235. The quadrature of a curve is the operation
of finding the area bounded in whole or in part by the curve.
If u denote the indefinite area limited by the curve, the
axis of Xf and any two ordinates, it was found (Art. 164) that

du = ydx z=zf(^x)dx;
y =/{x) being the equation of the curve referred to the reo-

tangular axis Oxj Oy.
^ If it is desired to have the

area limited on one side by the
fixed ordinate CAj correspond-
ing to the abscissa x = OA = a

5^ the integral must begin at a;=a/

and wo have

f{x)dx.

Finally, if the area is to be limited on the other side by the
ordinate BD, corresponding to a? = OB = 6, we have

u = SLre2kAGDB=zf /{x)dx.

When the coordinate axes are oblique, making with each
other the angle Â«, then

u = area A CDB = sin. w J f{x)dx.

392

M^

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393

M

N B OC

236. The definite integral is the limit of the sum, taken
between assigned limits, of an infinite number of infinitely
small areas (Art. 192). Observing that /(x) etc is equivalent
to /{x) AXj if we suppose ax =i dx io he positive, the element
/{x) A 05 will have the sign of/{x). Consequently the integral
will represent the difference between the sum of the segments
situated above the axis of x and the sum of the segments
situated below.

If, for example, the ordinate
changes, as in the figure, from
positive to negative, and then
from negative to positive, the
area between the ordinates AC,
BD, will be

Cf{x)dx = ACL - LMN+ NBD;

and if OL = h, 027"=: k, the sum of these segments will be
expressed by

f/{x)dx â€¢^j'^/{x)dx + fj{x)dx.

237. If y =/{x) is the equation of the curve CM, and
y J z= \p{x) that of the curve OM,
and the area bounded by these
curves and the ordinates AC,
BM, corresponding to a; = a,
a; = 6, is required, we have

Area C'CMM'= C f{x)dx - f\{x)dx

50

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394 INTEGRAL CALCULUS.

EXAMPLES,

Example 1. The family of parabolas is represented by the
equation y^^px"^, m and n being positive. We have

I m

du = ydx =zp^ x^ dx,
and

f*x I m , ^ 1 m-|-n ,

/X I TH â€¢71 * w>-T-w ,

p^x''dx = â€” - â€” p^x ** '

wnicn may be written

n ^ ^ n

IS

%^

V

p ^

u ~ â€” I â€” v x"" x^=^ â€” i â€” ^V'

But XT/ measures the area of the rec-
tangle OPMNj contained by the co-ordi-
nates of the point M, Hence, from the
above formula, we have

0PM I OPMN: : n : m + n,
0PM I OMN::n:m;

that is, the arc of the parabola divides the rectangle con-
structed on the co-ordinates of its extreme point into parts
having the ratio of n : m.

Reciprocally, the property just enunciated belongs to the
parabolas alone ; for the proportion

0P3I: 03IN::n:m
may be written

u\ xy â€” uil mm.

Hence (m -j- 7i) t^ = nxy, and, by diflferentiation, we have

(m -\-n)du:=^ nxdy -|- nydx ;

or, since du = ydx,

mydx = nxdy :

1 dx dy

whence m z=zn â€” .

X y

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Integrating

nly = rrdx -f (7, or Zy " = Ix"^ + (7,
putting Zp for (7, we have

ly^ = Zpa?"* or y* =zpx^
for the general equation of the curves which possess the
property in question.

For the ordinary parabola in which n i= 2, w = 1, we have

2 .

u=-xy.

Ex. 2. The hyperbolas referred to their asjnnptotes are rep-
resented by the equation x^y^=ip,
m and n being entire and positive
numbers.

Assume the asymptotes to be rec-
tangular, and let NCM be the branch
of the curve situated in the angle
xOy.

Suppose n^m, and let u = area AGMP, OA = a, OP=^x;
then

/x ^x 1 m

ydx = J p^ x*" dx,

V â–

Â£ A. P 00.

or

w = p^lx ^ â€” a ** V

As x increases, so also does u^ or the area AiJMP; and x and
w become infinite at the same time. If, however, we suppose
PMio be fixed, and a to decrease, the surface, while continu-
ally increasing, will remain finite ; and at the limit, when a = 0,

1 n â€” m

it reduces to Â«*â€¢ a? Â» . Hence the surface PMNL

n â€” m^

approaches a fixed limit as the point N approaches the

asymptote Oy.

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396 INTEGRAL CALCULVS,

This limit, which may be written xy, bears to the

rectangle PMBO the constant ratio of n to n â€” m: since,

denoting this limit by u, we have

n

n ^ m in II xy \ u =. xy.

^ n â€” m^

The converse of this is also true ; that is, no curves, ex-
cept those represented by the equation a3"*y*=^, possess
this property: for, from the preceding proportion, we have
u(7i â€” m) = nxy, which, diflFerentiated, gives

{n â€” m) rfw = nxdy + nydx;
from which, by substitution and reduction, we have

dx dy
X y

Integrating

rdy = â€” ndx -(- 0,

making C = Zp, then Zy** = Z -^ : hence a?"* y* =zp,

X

When m = n, the general equation takes the form xy =ip^
which is that of the equilateral hyperbola of the second

degree ; and we have

Â» , dx

y = ^,ydx=zp~,

X

and therefore u =zplx + C =^pl -

by making C =pl-. When^ = 1 and a = l,we have u = Ix;

and the area is then equal to the Napierian logarithm of the
abscissa.

Ex. 3. The equation of the cirqje, referred to its centre and
rectangular axes, is

x^ + y^=za^: .-.y = Va* â€” x^;

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597

and ydx = Va^ â€” x^ dx is the difiFerential of the area of a
segment limited by the axis of y and <f
an arbitrary ordinate PM, Denoting
this area by u, we have

= f y/a'-^x'^doi.
J

Hence (Ex. 2, p. 326)

u:=.-^x Va^ â€” (C* +

o" sm.""* -â€¢
2 a

Prom this we deduce the area of the sector OBM ; for

the area of the triangle OMP is measured by ~ a: ^a^ x^

which, subtracted from the expression for u^ gives

sector OBM=. â€” sin.~^ - = a ^ sin.""^ - = ^ arc MB:
2 a 2 a 2

that is, the area of a circular sector is measured by its aro
multiplied by one-half of the radius.

Ex. 4. If a and b denote the semi-axes of an ellipse, the
equation of the curve referred to its centre and axes is

Let u denote the area of a seg-
ment bounded by the axis of y and
any ordinate, as PM; then

a â€¢/
Describe a circle on 2a as a diame-
ter, and denote by u' the area of the
segment 5JfP0/ then

a

u':=zj^ Va2-a?Â»(fc,

(

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398 INTEGRAL CALCULUS.

b u b

t^' : w : : 1 : - ; or â€” , = - â€¢
a u' a

That is, the segment of the elUpse is, to the segment of the
circle which corresponds to the same abscissa, inthe constant
ratio of 6 to a ; and therefore, denoting the entire area of the
ellipse by A, and that of the circle by A'^ we have

A\A!\\b\a;
and, since A! = na^^ it follows that

^ = - na^ = Ttcib.
a

Hence the area of the ellipse is a mean proportional be-
tween the areas of two circles, having for diameters, the one
the transverse, and the other the conjugate, axis of the ellipse.

The ordinates PM, PN, are to each other as a to b, and
hence the triangles 0PM, OPN, are in the* same proportion ;

that is,

OPN _ PN _b u _b

OPM^PJfa' ^'^^u''^ a'

u - OP N _ b OCN_b^
u' - 0PM '^ a ^^ OEM" a'

and thus the area of the elliptical sector may be found in
terms of the area of the corresponding circular sector.

An ellipse may be divided into any number of equal sectors
when we know how to effect this division in a circle. It
would only bo necessary to describe a circle on the major
axis of the ellipse as a diameter, then divide the circle into
the required number of equal sectors, and through the points
in which the circumference is divided draw ordinates to the
major axis of the ellipse. The sectors formed by joining the
centre with the points in which these ordinates cut the ellipse
will be equal.

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Ex. 5. The equation of hyperbola is a^y'^ â€” J^^s _Â« __ ^252^

or y = - Vrc^ â€” a^ ; and the area of

the segment -d(JfP is expressed by
Hence (Ex. 6, p. 328)

Ex. 6. The differential equation of the cycloid (Art. 146) is

ydy

dx

=4

y^ dy=

Ir â€” y 4^^2ry â€” y^'

y'^dy

This integral may be found by Ex. 1, p. 370 : the following,
however, is a more simple process.

Put miz= 2r â€” y = z; then, denoting the area 0L2irM by
u', we have

ic' = fzdx=f{2r-y)dx:=f^2i^^^^'dy;
observing that the limits between which these integrals are

taken must correspond to z = 2r and a = 2r y. But

JV2ry â€” y^ dy is evidently the expression for the area of the
segment of a circle of which ti
r is the radius ; the segment
taking its origin at the ex-
tremity of a diameter, and
having y for its base. This L
segment is represented by
ADB. The area OiiV^ilf takes its origin from OL, and the cir-
cular segment from the point A, and both areas are zero when

X,

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400 INTEGRAL CALCULUS.

y =: : hence the constant of integration is 0, and we constantly

Wheny = 2r, the segment becomes the semicircle ^-0(7,

which is measured by -â€” . But

rectangle OLCA =OAx AC = nr X 2r = 2Â«r*;

' that is, the rectangle is twice the area of the generating cir-
cle : hence

3
area semi-cycloid OMCA z=L-^nr^^

and therefore the area bounded by a single branch of the
cycloid and its base is ihrte times the area of (he generating
circle; or, in other words, this area is Uiree-fourUia of the rec-
tangle having for its hose the circumference, and its aUityde
the diameter, of the generating circle.

Although the area of the cycloid may be said to be thus
represented by a part of a rectangle, it is not a quadrable
curve; for the base of the rectangle cannot be accurately
determined by geometrical processes.

238. Quadrature of curves referred to polar co-ordinates.

The differential of the area

PBM{kxt. 165) i&du=\ r^dd:

hence

the limits of this integral being
^T the values of d corresponding to
the points B and M.

Example 1. Applying this formula to the logarithmic spiral,
of which the equation is r = ae^^, we have

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Put PM =^ t\ and in the formula
make r =.r' ; then

""4m * * ' ~" 4m'

and v= 'J~y~''^y

The figure supposes PA to be the initial position of the
radius vector; that is, the position at which <9 = and
r = PA = a, and also that is positive when the motion of the
radius vector is in the direction of the motion of the hands of
a watch. Hence, when the generating point moves in the direc-
tion from A towards -B, d is negative. Let the motion take
place in this direction from the fixed radius vector PJR = r;
then, after an infinite number of revolutions, r' becomes 0,

and the expression for u reduces to u = â€”-.

2. When the length of the radius vector of a spiral is pro-
portional to the angle through which it has moved from its
initial position, its extremity describes the spiral of Archi-
medes. The equation of this spiral is r = ad; and hence
r =z a when 6 = 67^.29578 of the circumference of a circle to

For this spiral, we have

and, if the area begins when ^ = 0,
(7=0, and u = -a^dK When d = 27t,

D

u := area PAB = ^ a'^n^ is the area de-

scribed by the radius vector during the first revolution. In

61

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402 , INTEGRAL CALCULUS.

the second revolution, the radius vector again describes this
area, and also the diredkPBA'B' included between the first and
second spires. Hence the area PBA'B' is measured by

ia'j(4Â«)Â»-(2Â«)Â»j=^a^Â«'.

It is evident that during any, as the m***, revolution, the
radius vector describes the whole area out to the m^ spire,
and that, to find this area, the integral

must be taken between the limits (m -â€” l)2;r and 2mrtj which
will give for this area denoted by u"

u"=\a\m27iy-\ a''im â€” lY{27ty
o 6

In like manner, we have for the entire area denoted by u'^
out to the (m â€” 1)**^ spire,

u" â€” u' = ^a\27ty^m' - 2 {m- ly + {m - 2yL

which is the expression for the area included between the
(m â€” 1)*** and the m}^ spires.

If we suppose a = ^ , this formula becomes

u"- u'=^- 27r j m^- 2 (m - 1)3 + (w - 2y\

= ^^ ~ â€” !â€” ^^ â€”7t=: (mâ€” l)2;t;

o

and in this, making m = 2, we find 27r for the area included
between the 1^* and 2^ spires. Hence the area included be-

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tween the {m â€” 1)*^ and w*^ spires is w â€” 1 times that
included between the l^Vand 2^ spires.

239. The quadrature of curvilinear areas is sometimes
facilitated by transforming rectilinear into polar co-ordinates.

Take, for example, the folium of Descartes, which, referred
to rectangular axes, is represent-
ed by the equation

^' + y' â€” ^^"^y = 0-
This curve is composed of two
branches, infinite in extent, which
intersect at the origin of co-ordi- "^

nates, and which have for a com- ^

mon asymptote the straight line of which the equation is

To determine the area of any portion of this curve in terms
of the primitive co-ordinates, we must find what the integral of
ydx becomes when in it the value of y derived from the equa-
tion of the curve is substituted. This requires the^solution of
an equation of the third degree ; but if rectilinear be changed
into polar co-ordinates, the pole being at 0, there will be but
one value of the radius vector in any assumed direction ; for,
the origin being a double point, two values of r, each equal to
zero, must satisfy the polar equation of the curve, and the
first member of this equation must be divisible by r^

Ox being the polar axis, the transformed equation is
r^(cos.' -j- sin.' d) â€” ay'^ sin. cos. ^ = :
whenc3 a sin. d cos. d

siu.^ -f cos.^
For the area of the segment OMN, we have u = ^ l r^ddy

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404 INTEGRAL CALCULUS.

which, by substituting for r its value above, becomes

_ 1 r^ a^&in.^dcos.'^d , _a^ r9 Bin.^ 6dd

^'^2J ^{cos.^d + siu.^oy â– "2JoC08.*^(l.

+ ta.u}d)

Â»^\2

I tan.Â«^ ^

a^ I cos.^ d

To eflfect this integration, put

dd

l + tan.'^ = Â«; â€¢â€¢. ^2 = Stan.'^-

cos.*-*^'
and hence

1

tan/^

(l+tan.3^)

1

3(l + tan.'<9)

+ C:

a'
The area beginning when ^ = 0, Ave have (7 = -^ ; and con-
sequently

a^ tan.3 q
u =

6 1+ tan.'
The entire area OifX is found by making (? = - in the

a*
value of t^, which then becomes -77 ; for then the fraction

1.

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SECTION VI.

RECTIFICATION OF PLANE CURVES.

240. The rectification of a curve is the operation of find-
ing its length, and the curve is said to be rectifiable when this
length can be represented by a straight line.

Denoting by a the arc of a curve comprised between a fixed
point and an arbitrary point {x, y), we have

ds = s/dx' + d^2 ^ ^^ Ji ^ g! (Art. 161) ;
and, by integration.

By means of the equation of the curve, da may be expressed
in terms of either x ox y ; and, the integral being then taken
between the assigned limits, we have the length of the curve.

Example 1. The Common Parabola^ From the equa-
tion y^ = 2px of the curve, we find ydy =:pdx, dx = ^â€” ^.
This value of dxj substituted in the differential formula, gives

whence, making the arc begin at the vertex of the parabola,
by Ex. 5, page 327.

406

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406 INTEGRAL CALCULUS,

Since the integral is to be zero, for y = we have

By the substitution of this value of C, the formula becomes

Ex. 2. The JEllipse. From the equation of the curve

we get

dy _ h^x^
dx a'^y

hence

ds = dx\^l^E^ = dx Wf^\

in which e = is the eccentricity of the ellipse.

Suppose the arc C2Vto be estimated from the vertex O
of the minor axis ; ^ then, to get the
length of the arc CNA, the integral
of the expression for ds must be
taken between the limits x = and
x = a; but all the values of x
between and a will be given by
-"V â€” v X = a sin. <3p, the angle 9 varying be-
tween and - â€¢ The substitution in
the value of ds of these values of x and its differential gives

or

cfe = a Vi â€” c^ sin.*-* 9 d(p ;

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RECTIFICATION OF CURVES. 407

and therefore

â€¢/
This integral belongs to a class of functions for which we have
no expression except under the sign of integration ; and, to find
its approximate value, we must- have recourse to a series.
The Binomial Formula gives

^^^^r^'in^\ â€” 1 â€” -e^sin.^gj â€” ^ 7*

(1 â€” e^^m?(f) = 1 â€” -e^sin.^gj â€” ^ 7e*sin.*g)

113,.. llSSg.g

hence, for the arc CN, we have

8=:a(pâ€” ~ae^ f sin.' qpdgj "~ H t ^^* / Bin.* cpdcp

113
â€” :^ -^ ^ae^ f sin.^(pd(jp....

The integrals in the second member of this value of 8 may
be found by applying Formula 1 of Art. 221. We should thus

get, by taking all the integrals between the limits and-,

Â»5 / Â« Ijrt ll^lSrt

113 elS 5 Â«
246* 2462'

hence, for the arc C2fA, we have

2 (

\2 J 3\2 4 / 5V2 46 y
7\24 6 8 J )

This is a converging series, and the more rapidly so as e
becomes less, or as a and 6 approach equality. When the

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408 INTEGRAL CALCULUS.

eccentricity is very small, it would be safficient to compute

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