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I. F. Quinby Horatio Nelson Robinson.

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

. (page 20 of 22)

but a few of the terms of the series.

This value of the arc CNA may be found without using
Formula 1 of Art. 221 ; for, assuming the first equation in that

article, and taking the integral between the limits and - ,
we have

r* sin.** 9^9 = r* sin.'^-'gjdg).

In like manner,

If ^ ^

p Bin.â€” *ipdq, = ^^/* sin "-'qxiq),

TT

1 n

J* 8in*<pd<p = 22

/:

Multiplying these equations member by member, there
results

4 . â€ž. ^ (w-l)(m-3)(m~5)...3.1 ^
^ ^ w(mâ€” 2)(m â€” 4)...4.2 2

The values given by this, by making m equal to 2, 4, 6,...,
successively substituted in the value of Â«, lead to the result
before found.

The angle gt is found by the following construction : On the
major axis as a diameter describe a circle ; produce the ordi-
nate PN to meet the circumference at M, and draw OM; then

x=OP= OMcos.POM=z asm.BOM:
hence 9 = angle BOM.
Ex. 3. The Hyperbola. Assuming the equation

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RECTIFICATION OF CURVES, 409

of this curve, and proceeding as in the case of the ellipse, we
get

To simplify, put Va^ + 6'^ = oe ; then

"* "-^^ x^ - a"

Now, for one branch of the hyperbola, all admissible values
of X are comprised between + a and + oo , and for the other
branch such values are comprised between â€” a and â€” oo ; and
it is evident that all of these values will be given by the

equation x = by making op vary between and -^ for

cos. 9 ^ o ^ -^ 2

one branch, and between ^ and n for the other.

Substituting this value of ic, we have

, asin.g)c{g)

(tx â€” 5 â€” ,

cos/ 9

J , Va^e^ â€” a^cos.^qp , ae Jv cos.^op ,

and da = ^ Â»9 = r- ^1 r^ Â«9 â€¢

cos/ 9 ^ cos/9\ e*

whence (fig. Ex. 5, p. 399)
8 = AM=

J^ cos/g)\ 6*

Developing the radical in this integral, we get

1 cos.^g) 1 1 cos.^g)

1.1.3.5.. .(2n â€” 3) cos.2Â«9)) ,

Jo cos.2g)^ 2 e2 2 4 c*

2.4.6...2n e'

1 a
or ^ = aetan.g? â€” h "" ^ â€”

^ 6

a /â€¢^Cl 1 cos.^qp ,113 cos. qp , ) ,

62

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410

INTEGRAL CALCULUS.

The integration now depends on that of expressions of the
form cos.'^gdg), and may be effected by the application of For-

mula 1, Art. 221, after changing in it x into - â€” cp.

Ex. 4. The Cycloid. The differential equation of this
curve (Art. 146) is

2r~y

dx = dy \y. â€” , oTdy = dx 1-^

In the formula ds = s/dx^ -f- rf^', replacing dx by its value,
we get

ds = V2r(2r â€” yf^dy:

8 = â€” 2 V'2r(2r â€” y) + a

If 8 be estimated from the origin to the right, wÂ© must

have

and

8 = 0F= 4r â€” 2V2r(2r â€” y).
In this, making y = 2r, we
have 00' J the semi-arc of the
cycloid, equal to 4r, and the
whole arc therefore equal to
8r, or four times the diameter of the generating circle-
Estimating the arc from the vertex 0' to the left, then
(7=0, since at this point y =z2r ; and we have

O'P = â€” W'lr(2râ€”y).
But V2r(2r -y) = VG^ x 0C = PG:

hence arc 0'F=z2 chord FO; that is, the length of the arc
of a cycloid, estimiited from the vertex, is twice the corre-
sponding chord of the generating circle.

^ (

a *

tf

/L

d j

K

n^j

â– ^/Â»

V

%

d

3/

V

^ N

]

i * ]

B

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SECTION VII.

DOUBLE INTEGBATION. â€” TRIPLE INTEGRATION.

241. Double Integrals are expressions involving two
integrals with respect to diflferent variables. Suppose it is
required to find the value of u which will satisfy the equation

, , = q){x, y), the variables x and y being independent.

This equation may be written

d /du\ , ^ dv

d'lL
by making v = -^ - The function v must be such, that its

diflFerential co-eflBcient with respect to y, x being considered
as constant, is equal q){Xjy). We therefore have

v=f(f(x,y)dy = -Â£:

hence u must be such a function of x and y that its differen-
tial co-efficient with respect to x, y being constant, is equal to

f(p{x,y)dy; and therefore

The value of u is thus obtained by integrating the original
expression with respect to y, and then integrating the result
with respect to x.

The last equation is generally and more concisely written

^ =ff^i^j y)^^^yy or u=zff(jp{Xf y)dydx;

411

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412 INTEGRAL CALCULUS.

the first form indicating that the first integration is performed
with respect to x, and the second integration with respect to
y. The second form indicates that the order of integration is
reversed.

24:2. It was shown (Art. 91) that 3â€”7= j-^-, or that
^ ' dxdy dydx^

these partial differential co-efficients were the same, in which-
ever order, with respect to x and y, the differentiation is per-
formed. We will now prove that the result of the integration
in the one order can differ from that obtained in the other
only by the sum of two arbitrary functions, the one of a;, and
the other of y. Let u^ u^, be two functions of x and y, either

d'^u
of which satisfies the equation ;,â€” ,- = (p (x, y) ; then

d'^Ui d'^u^
dxdy dxdy

(dv
dx

= 0,

^"^ ^ \dy) "^ ^' P^^^^^g V = t^i - u^.

Now, ^ cannot be a function oi a;, otherwise its differential

co-efficient with respect to x could not be ; but it may be
any function of y. Hence we may put

^ = Ay) ; whence v = Jf{y)dy + x{x),
in which x{p^) denotes an arbitrary function of x. Putting
SAy)dy'='^{y)j'^{y) being as arbitrary sls /(y), we have

finally

v = Wi - U2 = rpiy) + Z(^);

as it was proposed to prove.

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DOUBLE INTEGRATION. 413

I ^(^; y)d^dy is the limit of

all the products of the form q){x, y)AXAy between the limits
of integration. Let q){x, y) be a function of x and y, which
remains finite and continuous for values of x between a and b,
and for values of y between a and /3.

To abbreviate, put (p(x, y) = z. Now, if we suppose x to
be constant while y varies between the limits a and j3, we
have (Art. 192)

j zdy =:]im. ^z Ay.

Multiplying both members of this equation by A a;, and sup-
posing X to vary between the limits a and b while y remains
constant, there results

//3
zdy = Uax lim. UzAy :
a

//?
zdy =:lim.2:Axlim.^z Ay =ilim.i:2;z AX Ay.

But lim. 2: AX J zdy = J J zdxdy

by the article above referred to : therefore

J J ^{^,y)dxdy = \\m.Zi:q}{x,y)AxAy.

Writers do not agree as to the notation for double integrals ;
some making the first sign J refer to the variable whose dif-
ferential comes first in the integral, while others make the
first sign J refer to the other variable. In what follows, the
first sign J will relate to the variable whose difierential is first
written in the indicated integral.

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414 INTEGRAL CALCULUS.

24:4:. In the last article, it was supposed that the variables
X and y were independent. It is sometimes the case, how-
ever, that the limits in the first integration are functions of

/b p3
I q){x, y)dxdy be

the required integral in which a = x{^)y ^^^ Â§ = V(^) 5 then

/ cp{x,y) dxdy = J ] ^ ^v{x,y) dxdy.

Suppose F{Xf y) to be the result obtained hj integrating,
first with respect to y, regarding x as constant ; then, for the
integral between the assigned limits for y, we have

F\x,rp{x)\ -F\x,x{x)\,
and finally

falxi^) "^ ^^' y^^^y = J^ (^ 1^' v^(^) I - ^ {^' x{^)\y^^'

When the limits of a double integral are constant, it is im-
material in what order, with respect to the variables, the
integration is efi'ected ; that is, a change in the order of in-
tegration does not require a change in the values of the limits.
But when the limits for one vai'iable are functions of the other
variable, and the order of integration is changed, a special
investigation is necessary to determine what tbe new limits
must be to preserve the equality of the results. A geometri-
cal illustration of this will be given in the next section.

245. Triple Integration. Let it be required to de-
termine a function u of the three independent variables x, y, z,

which will satisfy the equation j-zjâ€”f = J^* WÂ© ^^7 write

d^u d d^u __ pr

dxdydz dz dxdy

d^U , d d'^U , rrj

or -^ â€” - â€” ;- dz =:-j~ , -_- dz = Vaz ;

dxdydz dz dxdy

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rniPLE INTEGRATION. 415

hence by integratioa with respect to Â«?, regarding a? and y ^
constant,

T" being an arbitrary function of x and y. Again : we have

d'^u d du rxTj m>f
dxdy dy dx ^J '

which, by integrating with respect to y, x and z being conr
stant, gives

f^=fdySrd. + r + 8';

T' being an arbitrary function of x and y coming from jT^dy^

and S^ an arbitrary function of x and z.

Finally

u=zJ^dx = fdxfdyfVdz + T+ S+B;

Tj Bf and S being arbitrary functions, â€” the first of x and y

resulting from jT'dx, the second of x and z resulting from

fS'dx, and the third of y and z.

It is usual to write the diflFerentials together after the last
sign of integration : the above equation thus becomes

u =JJJ Vdxdydz + T+ 8+ B.

This example sufEces to show the manner of passing from
a differential co-efficient of any order of a function of several
variables back to the function itself. When the variables are
independent of each other, as has been here supposed, there

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416 INTEGRAL CALCULUS,

is no dependence between the arbitrary functions Ty 8y B;
but more commonly at the limits of the integral the variables
are not independent of each other. For example, the limits of
the integral with respect to z may correspond to z = F(Xy y),
z = Fi{Xj y) ; those with respect to y, to y =/{x)j y =/i(a;);
and, finally, those with respect to x, to a; = a, ic = 6.

By a demonstration similar to that given in the case of a
double integral (Art. 244), it may be shown that

/ dxj dyj (p{XjyyZ)dz = lim,2^2^Ii^x^yAz.

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SECTION VIII.

QUADRATURE OP CURVED SURFACES. â€” CUBATURE OF SOLIDS.

246* Let F{Xj y, 2) = be the equation of any surface
whatever, and take on this surface the point P, (x, y, 2), and
the adjacent point Q, {x -\- ax, y + Ay, % + AÂ«). Project these
points in P', Q', on the plane
X, y, and construct the rec-
tangle P'Q^ by drawing par-
allels to the axes Ox, Oy. The
lateral faces of the right prism
of which P^Q^ is the base will
intercept the element PQ of
the curved surface. Denote
by X the angle that the tangent
plane to the surface at the
point P makes with the plane
(Xj y). This plane is determined by the tangent lines drawn
to the curves Pq, Pp, at the point P. The tangent line to the

dz
dx

first curve makes with the axis of x an angle of which

is the tangent, and the tangent line to the second makes with

dz
the axis of y an angle of which -^ is the tangent. These

ay

are the angles which the traces of the plane of these two lines,

that is, of the tangent plane to the surface at the point P on

the planes {z,x), {z,y), make with the same axes. Now, from

63

417

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418 INTEGRAL CALCULUS.

propositions 1 and 3, chap, ix., Robinson's "Analytical Geome-
try," we readily find, without regard to sign,

1

cos.>l =

1'-(Â£)^(S1

The rectangle P^Q' is measured by AÂ« Ay, and is itiie pro-
jection on the plane {x, y) of the corresponding element of the

tangent plane. This element is measured by ^r heuce,

for the element of the tangent plane, we have
^x^y ( /dzy /dz\^)i

= seclAOJAy.
Let 8 denote any extent of the surface under considera-
tion, and assume that the limit of the sum of the terms
sec. lAxAjr, for all values of x and y between assigned limits,
is the area of the surface ; then

If the surface is limited by two planes parallel to the plane
(2, y) at the distances x=:a, x = b, and by the surfaces of
two right cylinders whose bases are represented by the equa-
tions y =z (p(^x)f y = '^{x)j we should have

and, when the cylindrical surfaces reduce to planes parallel to
the plane (Â«a?), (p{x) and yj{x) become constants c and e, and
the formula reduces to

*=/>/:i'+(S)V(fjr*

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VOLUHl^S OF SOLIDS,

419

247. A rea of Surfaces of Revolution. I f y =/(fl?)
be the equation of a curve referred to reetangulai* axes, the
differential co-effiiGix^nt of the area of the surface generated
by the revolTiition of this curve about the axis of x has been
foynd (Art. 167) to be

dS
dx

24:8^ Volumes of Solids. Consider the volume bounded
by the surface of W|hich Fi^x^y, Â«) = is the equation; and
t^rpugh the point jP, (a:, y, ^), in this surface, pass planes par-
allel to the planes (2, x), (z, y)] and also through the point
Q, {x + ^x, y + ^y, 2 + A2),
adjacent to tlie point P, pass
planes parallel to the same
co-ordinate planes. These
four planes are the lateral
boundaries of a prismatic col-
umn, having P'Q for its base,
and terminated above by the
element PQ oi the curved
surface. The volume of this
column is measured by zAXAy^
when AX, Ay, are decreased without limit; and the volÂ«me
bounded by any portion of the curved surface^ the plane
(Xj y) and planes parallel to the planes (2, ?/), (s, x), will be
the limit of the sum of a series of terms of which ztx^y is
the type. Denoting this volume by F, we have

^= ZzLxLy = J Jzdxdy,

i

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420 INTEGRAL CALCULUS.

Prom the equation F{Xj y, z) = 0, which is the equation of
the surface, we have z=zq)(^x, y). If we integrate first with
respect to y, we get the sum of the columns forming a layer,
included between two planes perpendicular to the axis of x;
and hence the limits of integration with respect to y become
functions of x, and we should have Jzdy =/{x) ; /{x) being,
in fact, the area of the section of the solid made by a plane
parallel to the plane {z, y). Thence, finally, Â¥=- Jf{x)dx.

249 â€¢ Volumes of Solids of Revolution. The differ-
ential co-efl5cient of the volume generated by the revolution,
about the axis of x, of the plane area bounded on the one
side by the axis of x, and on the other by the curve havmg
y '=f(x) for its equation, has been found (Art. 166) to be

- = ny-=f(^^):

hence, by integration,

r=nfy''dx = ff{x)dx.

Here, as was the case at the end of the last article, f(x) = ny^
is the area of a section of the solid made by a plane perpen-
dicular to the axis of a;; and the integral is the expression for
the sum of the elementary slices into which we may conceive
the solid to be divided by such planes.

APPLICATIONS.

Example 1, Required the measure
of the zone generated by the revolu-
tion of the arc MM* of a circle about
P Vlk. X the diameter BA, The equation of the
circle is a?* -f- y* = R'^. Denoting the area 6f the zone by 8,
ifOP=a,Or=^ 6, we shaU have (Art. 247)

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EXAMPLES.

421

= 2;tJ2 (6-a) = 27tR x PP'.

To get the entire surface of the sphere, the integral must be
taken between the limits a; = â€” iJ, a? = iJ, which will give

8=4:rtB\

Ex. 2. Suppose the ellipse of which
the arc BMA is a quadrant to revolve b
about its transverse axis : required the
measure of the surface generated by
the portion BM of this arc, beginning
at the extremity of the conjugate axis. We now have

Prom the equation of the ellipse, a^y^ + 6^a;^ = a^fe^, we

dy b^x

e-et rp- := â€” â€”r- ; whence
^ dx a^y^

A^

\ ^\dx) "" a'^y - a^y '

and finally, by making s/a"^ â€” 6^ _- ^^^ ^^ have

^hm='-^

ay

therefore

â€ž ^ b r^ , 2;r66 /â€¢* \a^ â€ž ,

S=27t- \/a^^e'^x^dx=zâ€”- J:^ ~^'^-

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422 INTEQtUt CALCULUS.

But (Ex. 2, p. 326)

Ne^ 2 Ne2 ^ 2 e^ a

therefore

â€¢ Tihe I la- 2 1^'^- -1 ^\

â€¢If, in this expression, we make a; = a, and take twice the
iQSult, we get

^=2;r62 + ?^sin-^e

for the entire surface of the prolate ellipsoid of revokition.

Suppose, now, that a <^ i, or that the e^llipse is revolved
about its conjugate axis, and put Vfr'^ â€” a* ==&e; then we
shall have

But (Ex. 5, p. 327)

therefore

a- /

^^.+^'+M'*-l^'h<'-

Since this integral should be zero, for a; = we have

^"" "a^ 62^2 Zftg- e ^fee*

hence

\

a"
6e

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CUBATUEE OF SOLIDS. 423

If in this we make x = a, ariid take twice the result, wo shall
have

8= 2.6 Va-^+W^ + ^ I ^ + Vf+b^^

for the entire surface of the oblate ellipsoid of revolution.
If we suf^ose a^^b, and therefore e = 0, the second term

in the last expression for 8 takes ihe form -; bwt, by the nile

for the evaluation of indeterminate forms, we readily find

lim.A ? /^i:

whence we have 4fra^ for the surface of the sphere.
Ex. 3. Cubature of the JEllipsoid of Revolution.

The equation of the ellipse, referred to its jnajor axis and

the tangent line at its vertex, is y'^ :::=: â€” (2ax â€” x^)] and there-

a"

fore, for the volume of the ellipsoid, we have (Art. 249)

V^â€”i (2ax~x'')dx=z^lax'' ).

a- */ a^ \ "6 /

To get the entire volume, we make x:=2a; and then

â– =fe

r= '-^ (^a^-\ a') == \^7tb''a.

This is the volume of the prolate ellipsoid. To get that of

the oblate ellipsoid, a and b must be interchanged in the last

formula. We thus get, for the measure of the entire volume,

4

-na^b; from which it is seen that this volume is greater than

o

the first. Making a = b, the ellipsoid becomes a sphere, the

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424 INTEGRAL CALCULUS.

4
volume of wliich is expressed by ^^ct'; and, for the volume

of a spherical segment of a single base, the expression is

n

-a;^(3a â€” x).

Ex. 4. Volume generated by the Revolution of a

In the formula F=f7t7/^dxj substitute for dx its value

tidtj
dx = ^-^ 7 derived from the equation of the cycloid.

and we have

J {2ry-y^f
but (Ex. 1, page 370)

J{2ry-y'Y 3 3 J {2ry - y^f

f-^^^=-h2ry-y^)^ + '-rf-y^,
J{2ry-y^y 2 2 J(2ry-y'f

f-y^ = - i2ry - y^)' + r fâ€”^
J(2ry-y^Y J {2ry - y^

= â€” (2ry â€” y') + r ver.sin."' - â€¢
therefore, by substitution and reduction,

V= C-^^y^ = -ni2ry-rHyl +^-ry+^-r^\
J(2rv-v^Y \3 6 2 /

+ 5Â«r'ver.sin.->?^ + C.
2 r

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EXAMPLES,

425

Taking this integral between the limits y = 0, y = 2r, and
doubling the result, we have, for the entire volume generated
by the revolution of a single branch of the cycloid,

Ex. 5. Volume of an JEllipsoid. Take for the co-
ordinate axes the jyrincipal axes of the ellipsoid. The equation

x^ y^ z^
of its surface is then â€” Â« + r^ +

= 1.

The section PMM' of the ellipsoid made by a plane parallel
to the plane ZOy^ and at the dis-
tance OP == X from the origin, has
for its equation

t

6^ â€¢ c a"

The semi-axes of this section will
be found by making in succession
a z= 0, y = ; they afe

i.^' = gX-i!,â„¢=oJ.-i:

hence the area of the section is

^â€¢2^ nbc

Ttbc

(â€¢-\$)=?(Â«â€¢-")>

and, for the volume of the segment included between the
planes ZOy and PMM'^ we have

V=

= â€”7 1 (a^ â€” a?^)c?a3 = â€” - (a^x-â€” -â€” ) â€¢

To get the volume of half the ellipsoid, make in this formula

2
cc = a, which gives F= ^nabc; and hence the entire volume

4
is measured by ^ ;ra6c.

64

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426

INTEGRAL CAICULUS,

Ex. 6. The areas of surfaces an4 volunj.es of solids have
thus far been found by single integration. As an example of
double integration, let it bo required to find the volume
bounded by the surface determined by the equation xy =: az,
and by the four planes having for their equations

x = x^, x=:x^j y=yu y â€” Vi'
The expression for this volume is

1

4a

(^., - x^) (?/2 â€” y;) (x^y^ -f x,^y.^ + x^y., + x^y{)

= 7 (-2 - â€¢^l)(y2 â€” y i)(m + 2-2 + ^3 + 24),

in which 2,, s^,, z^, 24, are the ordinates of the points in which

the lateral cd;res cf tlie volume considered pierce the surface

xy 1= az,

IZx. 7. To illustrate triple integration geometrically, in tJia

figure suppose planes to be
passed perpendicular to the
axis of z. Let two of these
planes bo at the distances 2
and z-\- Lz respectively from
the origin of co-ordinates;
cutting from the elementary
column JPQ' a rectangular
parellelopipedon cib measured
by LxLyLz. This parallel-
opipedon may be considered

as an element of the whole volume V: hence

V^^JJfdxdydz.

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EXAMPLES. 427

Required the portion of thÂ© volume of the right cylinder
that is intercepted by the planes z =â–  a; tan,^, z = a? tan.^%* the
equation of the base of the cyliuder being a?'^ + y'^ â€” 2ax = 0-
Here the limits of the integral are z =^xtain.O, z =^xia.n.d^j
y = â€” >/'lax â€” x^, y z= -\- \^2ax â€” x\ a? = 0, a; = 2a ; there-
fore, denoting the values of y by â€” y^, + Vu

r'' f dxdydz

t/ â€” Vy J X Uii.^

J _ (tan. 6' â€” tan. 6) xdxdy

/2a
^ x\/2ax'-x'dx

= (tan.^' â€” tan. ^)7ra*.
The base of this cylinder is a circle in the plane {x, y) tan-
gent to the axis of y at the origin of co-ordinates ; and the
secant planes pasB through the ^^^^ ^^_ ^^^^p,^

origin, and are perpendicular ^^HB *^5^irf ^^^T^L^
to the plane (2, x). The re- ^^/^ft h^k^ ^
quired volume is therefore the ^^^^J^ *^^i ^^ ^B
portion of the cylinder included L'*^! â– ^LJk ^^^H
between the sections OP, 0P\ ^j^^^^^'^^i^ft^ .^^m
It can be seen from this exam- |^3^^^^|C ^ Ir^^
pie why, as was observed in ^^^^^bKKK^ mr M W
Art. 244, when there is a rela- fc..l^FW^Zji^*'^ /^^WJI
tion between the variables, at

the limits of an integral, the order of integration canaot be
changed without at the same time ascertaining if it be not
necessary to make a corresponding change in the limiting
values of the variables. la this case, after integrating with
respect to 2, we integrate with respect to y, taking tlie inte-
gral between the limits y = â€” (2aa3 â€” x^) , y = + (2aic â€” x^)'^

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42S INTEGRAL CALCULUS,

that is, the integral is considered as bounded by the circum-
ference of a circle tangent to the axis of y at the origin ; but
by what portion of the circumference is not specified until the
h'miting values of x are assigned. The integral with respect
to X is then taken from a; = to a? = 2a, which thus embraces
the whole circumference. i

l>ut it is obvious, that, if the order of integration with respecfc
to X and y be reversed, then, that the integral may embrace
the whole base of the cylinder, the limits with respect to x
must be a; = a â€” Va^ â€” y'-*, a;=:a-{- V^'* â€” y^; and those
witli respect to y must be y = â€” a, y=:-\-a. We now have,
denoting the limiting values of a? by aji, â€” a?i,

{ ' ^ dydxdz

= ( r_' ( ^^"- ^' â€” *^^- ^) ^dydx

= ^^J_ (tan.^' - ia.u.d)Va^â€”y^dy

= (tan.<9^ â€” tan.(?)7ra' (Ex. 2, page 326) ;
which agrees with the first result.

230* Polar Formula. Th'e polar equation of a plane
curve being r = gp(<9), if 8 denote the length of an arc of the
curve estimated from a fixed point, the differential co-eflScient
of this arc (Art. 163) is

'â– ' + (t)V<Â» (1):

or, by taking r as the independent variable.

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POLAR FORMULAE. 429

Example 1. Applying Formula 1 to the spiral of Archiine-.
des, the equation of which is r = aO^ we have

= 1 (i + ^^)* + ^^{^ + (i + ^^)*} + c^.

If the arc considered begins at the pole where ^ = 0, then

Ex. 2. For the logarithmic spiral, we have r =. ba^ , or

- 1 r dd c

r = be^ by making a = e*" ; . â€¢ . (? = d =-, and -^ = - : whence^

by Formula 2,

8z=f^{l+c')dr=:z^{l+c')r+a

If the limits of the integral correspond to the radii vectores
Tq, Vif the length of the arc is

Since r â€” \9> the expression for the tangent of the angle

made by the radius vector of the curve at any point and the
tangent line at that point, we have, calling this angle cc,

ds
tan.a=:c; hence sec. a = ^(1 + c^), and â€” = sec. a; there-
fore 5 = r sec. a + (7, and the definite portion of the arc an-
swering to Vqj Vi , is (ri â€” ^o) sec. a.

2S1. To find the length of a curve in terms of the radius
vector and the perpendicular demitted from the pole to the

dr
tangent line to the curve at any point, we have cos. a = â€”

ds

(cor. Art. 163) : hence, if p denotes the length of the perpen-
dicular,

r r da r

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430

INTEGRAL CALCULUS.

therefore

ds
dr

/rdr

252. The length of a curve may also bo expressed in
terms of the perpendicular and its inclination to the initial
line.

Let X and y b^ the co-ordinates of any point M of the curve,

and denote by 8 the length
of the curve included be-
tween the fixed point A
and the point Jf. From
the origin iei, fall the per-
pendicular OP upon the
tangent to the curve at the
^ point If, and make OP =i?,
MF = u, and the angle POx = d; then, from the figure, we