u=zx sin. 6 — y cos. d.
Also we have
ds
-4 = — cot. d, -r- = — cosec. Q :
dx MX
therefore
dp
~dl
dx
= —X sin. d '\-y cos. e + COS. ^ ;^ + sin. d
dd
dy
do
dy __
But, since id the independent variable, T^ ~* "^ ^^*- ^
dy
dd COS.
do
mav
be written -:Tr = —
sin. e '
dy
whence
dx
sm. fif -V- + COS. ^ -,- = :
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POLAR FOBMUUB. 431
dp
y- = —'X sm. 6 -\-y cos. 0= ^u,
dd
^ = - =-a.coB.^-ysm.^^sin.^^ + cos.^^^.
dy
But, from -p- = — cot. 0, we get
ax
dy ^ dx
COS. ^ -^ = — oos.^^ co8ec. ^ -t; :
do do
dx ^ dy . dx ,^ ^rfaJ
.*. — sm.(?^r- +cos.^-^= — sm. ^^;- — cos.*^co8ec.^^7
do ^ do do dd
dx/s\n.^O + cos.^O\
zs: *— — ( I m — (
do \ sin. I
dx /sin.2 + cos.^ 0\ dx
' ^ — cosec. - - •
do
fjo fijt dx
The equation -— =2 — cosec. ^ gives 3- = cosec. -j- ; hende,
dx do dO
by substitution,
^V ^ • ^ , ^
cf^^ ^ ^ do
^ds
="^ + do''
therefore
or 8'{-U=z JpdO.
Taking the integral between the limits 0^, 0^ Sq, «j, Wq; }^i)
being the corresponding values of 8 and Uj we have
The sign of u will be positive or negative according as the
angle POx = ^ is greater or less than the angle MOx. These
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432
JJSITEGRAL CALCULUS.
results may be used for several purposes, the most important
of which are, —
First, To find the length of any portion of a curve, the
equation of the curve being given. In this case, from the
equation of the curve and the equation -~ =z — cot.^, x and y,
and therefore p =^x cos. -{-y sin. 0, can be determined in
terms of 0; and, by integration, 8 may be found from the
equation » z= - ^ -(- JpdO.
Second^ To find a curve, the length of a portion of which
shall represent a proposed integral. Here, if the integral be
JpdO, p being a function of 0, the equation of the curve is
found by eliminating between the equations
dp . . dp
X =p COS. 6 — fs sm.^, y =p sm. -\ - ^ cos. 0,
hich we get from the equations
p=^x cos.^ -\- y sin.^, -^=i — x sin. 6 + y cos. d.
do
dp
The proposed integral will then be represented hy s — -^
APPLICATION.
Let BMO be a quadrant
of an ellipse of which the
equation referred to its
centre and axes is
a^y^ + b^x^=zan\
This equation, by making
P = a2(l — e^)f may be
put under the form
2=:(l-c«)(a«-a?^).
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POLAR FORMULJS, 433
Make POx = 0.
I'hen, from the properties of this curve, we have
0T2 = -' tan.«PrO==cot.2(? = lli:i^'.
x^ a* — a;'
From the last equation, we get
a^ — rr2 , x^l\ — e^)
sin/d ;= -2 2—=-, cos/ ^ = -^^ ^—^ :
W =Wt' X cos.'<? = a*^ii^>,
or â– OP = y = a ^ ^/_ ^ , ^/ = a V 1 ~ 6^ sin.^ <?.
Therefore
Caf + MP = s + u = aJVl — e^Qm.^d dd.
It is here supposed that the integral takes its origin at C>
the vertex of the transverse axis. Now, if the point A be so
taken that the angle BOA = d, it has been shown (Art. 240)
that
Arc BA = aJVl — e^ BiiL^ddO :
CM+MP = BA.
Also we have
Tim ^P ^^* sin. COS.
MP = — ^= , . :
do Vl-e^sin.'/?
and, X being tlie abscissa of the point M^
dp .
x = p cos. 6 — ^ sm. d
^ do
=ra(l— e^siD.2^)*cos.^-
ae^ sni.^ cos. a cos. 6
C5
(1 -c2sin.2(?)* (1 — e«sin.»^)*
Digitized by VjOOQIC
43 i LXTEGIIAL CALCULUS.
Thereforo MP = e^x sin. 0; and, x' being the abscissa of A,
WO have x' = a siu. 0: . • . MP =■- - - , aild hence
a
a result known as Fagnani^s Theorem.
From the values of x and x% we get
t
a^ — cc'
1 — e- sin.'-^ ^
1
which gives
e^ x' x' — a- {X' + x') + a*=:0,
an equation whicli is symmetrical with respect to a? end ic':
hence, if wo have
we also have
BA- CM=-xx',
a
BM - CA~xxr
a
253. Curves of Trouble Curvature. A curve of
double curvature is one, three of the consecutive elements
of which do not lie in the same plane. Such a curve must
be referred to three co-ordinate axes, and requires for its
expression two equations which represent the projections of
the curve on two of the co-ordinate planes.
Let the equations of the curve be
y=J\x) (1), z = ^{x) (2);
(1) being the equation of the projection on the plane (x, y),
and (2) the equation of the projection on the plane (x, z). If
X, y, 2, are the co-ordinates of a point of the curve, and
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CURVES OF DOUBLE CURVATURE. 435
aj + Aa:, y -4- Ay? is + az, the co-ordinates of an adjacent point,
"then, by the principles of solid geometry, the length of the
chord connecting these points is
{(Aa;)» + {Ay)'- + (A2)'j*.
Then, if s is the length of to arc of the curve estimated from
a fixed point up to the point {x, y, 2), that of the arc from the
same fixed point up to the point (x -\- ax, y + Ay, z -\- a«)
will be expressed by b-^- ls. We shall assume
,. AS
lim
|(Aa;)' + (Ay)^-H(Az)«p
and therefore
dx
18
_y A^ __. J
The two equations of the curve enable us to express -~ , -^~,
in terms of x; and, by integrating, 8 will then be known in
terms of x.
Any one of the three variables may be taken as independ-
ent ; and the above formula may be changed into
or
Digitized by VjOOQIC
436 IXTEGRAL CALCULUS,
When Xj y, and z are each a known function of an auxiliary
variable, ty as may be the case, then
dy
dy_ dt ^
dx dx '
dz
dz dt ,
dx dx
dt
Jt
and we may have
or
or
=/h(l)'+(D]'-.
â– =/l(SJ+(l)'+(l)T-
254:. To convert the formulas of the last article into polar
formulae, take the pole at the origin of co-ordinates, and denote
by the angle that the radius vector makes with the axis of
2, and by g) the angle that its projection on the plane {x, y)
makes with the axis of x ; then we have the relations
x^=zr sin. 6 cos. q, y z=zr sin. d sin. qp, z=^t cos. 6.
These three equations, together with the two equations of
the curve, make five between which we may conceive r and qp
to be eliminated, leaving thr^e equations between x, y, z, and
: hence, x, y, and z may be regarded as known functions of 0.
Therefore
dx , dr , ^ . dw
-T- = sm. 6 COS. w — r sin. d sin. op -r- + ^ cos. 6 cos. qp,
do ^ do ^ dd^ ^'
dy . . dr ^ , ^ cZqp .
-r- sm. sin. 9 -^ + ^ sm. cos. 9 -^ + ^ cos. sm. op,
do do ad
dz dr ,
- = COS. -^ r sm. :
do do
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POLAR FORMULM.
437
••(IJ+(S)'+(D'
-|- r^, and
angiag th(
+ ^^ sin.2 '
which, by changiag the independent variable, may become
+i+.-.i„...(|)]'*,
or
=/lK0+(lJ+""-f^-
<855. Polar Formula for Plane Areas. In the
curve BMQ of which the polar
equation is r = g)(^), let r, ^, be
the co-ordinates of the point M, and
denote by A the area bounded by
the curve, the radius vector PB
drawn to the fixed point B, and the
radius vector PM, Then (Art. 165)
dA
dd
Let \\j{d) be the function having cp{0) for its differential
co-efficient; then A=z\f)(d)-\- O: and if -^i, -^2, denote the
areas corresponding to the values ^i, ^27 of the vectorial angle,
we have
A,=xp{d,) + C, A, = xp{d,) + C;
.-. A, - A,z=xt^{d,)^y^{d,)==lj'j^\cp{d)\'dd.
• = ^|qp(^)j^ r.A = lf\<p{0)\'dd.
Digitized by VjOOQIC
438
INTEGRAL CALCULUS.
Example 1. For the parabola, when the
pole is at the focus, and the variable angle,
measured from the axis, begins at the ver-
|o tex, we have x =p — r cos. 0, y =^r sin. ;
from which, and the equation y* = 2px of
the curve, we get
_ p 1 p p^ i' do
'~ 1 + COS. 0~2 ^' **• ~~
COB. -
p^ [* de
I cos.<-
dO = ^X.u.- + - i.n.^-^G;
= ^/(l + tan.^^)se
...^.-^. = ^'(tan.^-tan4)+g(tan4^-tan.3^).
n
p-
Making (9i = 0, d^^ =-^, we have for the area ^ + 4i^? or ^'
Ex. 2. The equation of the logarithmic spiral being r z=be^ f
we find
1 /» 2(? J2^ 20
A = ^ fb'eTdd=i ^e^ + G,
^56* A polar
formula involving
double integration
may also be con-
structed for plane
areas. Suppose the
area included be-
tween the curves
BMEj bmCf and the
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POLAR FORMULAE, 489
radii vectorcs PB, PE, is reqaired. Divide the area up into
curvilinear quadrilaterals by drawing a series of radii vectores,
and describing a series of circles with the pole as a centre.
Let no be one of these quadrilaterals, and denote the co-ordi-
nates of n by r, 6; and of o by r + Ar, ^ -{- A^, Now, the area
no is the dilBFerence between two circular sectors; and the
accurate expression for this difference is rArA^-|- -(Ar)^A<?,
the ratio of the second term of which to the first is
rArAd 2r
This ratio diminishes as Ar diminishes, and vanishes when
Ar:=0: therefore we may take rArAd as the expression for
the elementary area, since, in comparison with it, the neg-
lected term -(Ar)^A^ ultimately vanishes.
257* In the last article, it was shown that tavaO might be
taken as the expression for the polar element of a plane area.
If we suppose this area to be the section of a solid by the
plane (a?, y), the column perpendicular to this plane, standing
on the element vataO as a base, may be regarded as an ele-
ment of the solid. The volume of this column is measured
by zrArAd ; and therefore, for the volume V of the solid, we
have V= ffzrdrdO.
The value of a as a function of r and will be given by
the equation of the surface bounding the solid.
Example. !^equired the measure of the volume bounded
by the plane (x, y), and the surfaces having
a:2-j.y2_^2 = (1), a;» + y«-26x = (2),
for their respective equations.
Digitized by VjOOQIC
440 INTEGRAL CALCULUS.
Denoting the polar co-ordinates of a point in the plane
{x, y) by r and dy 6 being measured from the axis of a?, we
have
a?z=rcos.<9, y = rsin.(? (3);
therefore a?' -J- y^ = r*, which, combined with (1), gives
r^z=:az: . • . 2 z= — .
a
From (2) and (3) we find r = 2bcoa,0 : hence, for this exam-
ple, we have
V = JJzrdrdd = J J- drdO.
To embrace the entire volume comprised between the sur-
faces indicated, the integral must first be taken, with respect
to r, between the limits r = 0, r = 26cos.(?, since is assumed
as the independent variable ; and then the integral of the
result must be taken between the limits ^ = h > d =z — - .
Thus
1 26C05. B ^ **
"J.
2
COQ^ddd
IT
^*'* cos.'dd0 = l-7t (Art. 221).
258 • Suppose the polar element r^r^d of a plane area to
revolve through the angle 2;t, around the fixed line from
which the angle is estimated. A solid ring will thus be
generated, the measure of which is 27rrsin.^rArA^; since, in
this revolution, the point whose polar co-ordinates are r, <?,
Digitized by VjOOQIC
POLAR FORMULA. 441
will describe a circumference having rs'm.d for itd radius.
Denote by (p the angle which the plane of the generating
element in any position makes with its initial position ; then
9 4" ^9 "^^U ^^ ^^^ angle which the element in its consecutive
position makes with the initial plane. That part of the whole
solid ring which is included between the generating element
in these two positions is measured by
(cp + Aqp) r^&m.dArAd — (pr^ sin.OArAd =. r^ sin.dArAdAcp.
This may be assumed as the expression, in terms of polar
co-ordinates, for an element of the solid : hence, for the vol-
ume Vof the whole solid, we have
V=: fj fr^ sin. Odrdddcp,
in which the limits of integration must be so determined from
imposed conditions, that the integral may embrace the entire
solid to be found.
Example. Required the volume of a tri-rectangular pyra-
mid in a sphere. Integrating the above formula, with respect
to r, between the limits r = 0, r = a, a being the radius of
the sphere, we find
V= f f fr^ sin.ddrdddx = f f^ sin, ddddcp.
Now, a^ain.dAdAqi is an element of the spherical surface;
and o- sin.dAdAcp is therefore the expression for an elementary
spherical pyramid having a^sin.^A^Aqp for its base. By this
first integration, therefore, the element of the volume has
changed from an element of the solid ring, generated by the
revolution of tataO, to an elementary spherical pyramid.
56
Digitized by VjOOQIC
442 INTEGRAL CALCULUS.
IntegratiDg next, with respect to d, between the limits
6 = 0,6="^,
we have
since
JsinMd = — cos.<9 : . * . J^ sin.^d^ = 1 .
2
By this second integration, the elementary volume has
become a semi-ungula, or a spherical pyramid, having a bi-
rectangular triangle for its base; the Vertical angle of the
triangle being Acp,
We finally integrate, with respect to cp, from qp = to qp = ^i
and get for our result
Digitized by VjOOQIC
SECTION IX.
DIFFERENTIATION AND INTEGRATION UNDER THE SIGN /. — EULE-
RUN INTEGRALS. — DETERMINATION OP DEFINITE INTEGRALS
BY DIFFERENTIATION, AND BY INTEGRATION UNDER THE SIGN /.
259. Whatever function of x, f{x) may be, there exists
another function, (p{x), of aj, such that (f^{x)=:f{x)\ and
therefore ( f{x)dx = q:.{x) -{- G (Art. 191), G being an arbi-
trary constant.
Denoting by w the integral of f{x)dx, taken between the
limits a and 6, we have
u =1
= r f(x)dx = (p{b) — g}(a).
The definite integral u is independent of a?, but is a func-
tion of the limits a and b; and its diflferential co-eflBcient with
respect to either of these limits may be obtained without
effecting the integration. For, since
u = (p{b) — (p{a),
we have
du ^, . du ,,,v
f^nd, because q)'{x) =/(x),
du .. . du .,,.
du = /{b)db—/{a)(h.
Hi
(
Digitized by VjOOQIC
444
INTEGRAL CALCULUS.
II
/
^'
ILLUSTRATION.
Lety =/(a?) be the equation of the curve JOT referred to
the rectangular axes Ox, Oy, If a
and b are the abscissad of the points
f{x)dx will repre-
sent the area AMNB. Give to a and
h the increments
A A'
B B' X
then
AU
La
AMiTA' Lu BNN'B'
A A'
Ab
BB'
The definite area AMNB is obviously a decreasing func-
tion of the first limit a, and an increasing function of the
second limit b : therefore
,. AU ,. BNN'B' .,,,
lim.-=lim.-^^^=/(6).
Regarding the areas A MM' A', BNN'B', as elementary, we
see that the total increment of the area AMNB is the differ-
ence of the increments that it receives at the limits.
260* Suppose /(x) to contain a quantity, t, independent of
f{x)dx with re-
a
spect to t is reouired. Replacing /(a;) by f{x, t), we have
f{x, t) dx.
If the limits a and 6 are independent of / we have by giv-
Digitized by VjOOQIC
DIFFERENTIATION UNDER THE SIGN f. 445
ing to t the increment A< {^u being the corresponding incre-
ment of u),
/b pb'
/{Xj t-\-At)dx^ j /{x, t)dx
'■"fa (•^^'^' ^ + ""^^ ""-^^^^ ^^y^"'
^i J a ^t
Now, by Art. 15, we may write
f{x,t + M)^f{x,t) _ df{x,t)
M "~ dt '^^'
in which 7 is a quantity that vanishes when At vanishes. De-
noting by f the greatest of the values of 7, we have, gener-
ally,
f ydx <^ (b - a)f ;
and, when neither a nor 6 is infinite, ^6— a)fj and therefore
/* . .
ydx, will ultimately vanish :
A^ dt J a dt
APPLICATION.
Resuming the formula
just established, suppose g)(a;,<) to be the function of which
/{Xf t) is the differential co-efEcient with respect to a?, and
df(x t)
Vf{x,t) to be the function of which '^ ^ ' ^ is the differential
(ZC
co-eflBicient with respect to x; then (1) becomes
^-^-H'^^ = Hh,t)-Ha,i) (2).
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446 INTEGRAL CALCULUS,
If /(a), t) and a are both independent of 6, (2) may bo written
^^M^C = Hb,t) (3);
(7 denoting the sum of the terms which are independent of 6.
Since we may give to b in (3) any value we please, replace h
by x; then (3) becomes
V.(x,0 = ^;|^ + ^ (4).
Dropping the constant (7, which may be restored when neces-
sary, and putting for the other terms of (4) their equivalents,
we have
1 dx
Example. Let /{x, t) = jz^^t^t , then f{x, t)dx = j^p^^, :
J/ix,t)dx=J^-^^ = U^n.-Hx,
and - f/(x,t)dx = -(^tnn-Hx\= f^^l^ dx
dtJ-'^ ' ' dt\t ) J dt
""J It (hh^) ^=~i {i^t^x^ ^''-
/dx
^ ^ , we find, by differentia-
tion, that of the more complex integral J .^ ^ dx.
/b
f{x, t)dxj both a and h
are functions of /, then -.- will consist of three terms ; since
Oft
in this case, to obtain the total diiFerential of u, we must dif
ferentiate it with respect to t, and also with respect to both
Digitized by VjOOQIC
DIFFERENTIATION UNDER THE SIGN f.
447
a and h regarded as functions of /, and take the sum of the,
results. Thus we should have
du
du db du, da
db dt da dt
Under the above suppositions, the second and higher differ-
ential co-efficients of u with respect to t may be found. Thus,
by differentiating each of the terms of the last formula with
respect to ^, we get
d'u_f^d'f{x,t)
+/(&,<) dti^—dir\^) + ^ —dT -dt
f.f.d'a d/{a,t) /daV df(a,
t) da
~ dr
ILLUSTRATION.
Let y =/(Xj t) be the equation of the curve CD referred to
the rectangular axes Ox, Oy, and ,
y =/{Xj t'\' At) that of the curve *
JEF. Put
OM=a, ON=b,
MM'zzLLa, NN' = Ab.
/b
/(Xj t) dx denotes
a
the area MNDC, and u -{- au the
area M'N'FE:
Au = EE'F'F+DNN'F' - MM'E'O,
Lu_ EE'FF DNN'F' MM'E' (7
A^ ~~ A^ At At
M M
Digitized by VjOOQIC
448 INTEGRAL CALCULUS,
It 13 plain that the first term in this value of -— is the ratio
of A< to the increment of the area due to the change' from
the curve CD to the curve EF. The limit of this ratio is the
limit of C /(^>< + ^0-/(^>0 ^^ Sq a,g ^^^ ij^^jt ^f the
J a At ' '
second term is the limit of /(6, t) — - , and the limit of the third
Ar
term is the limit of /(a, t) — - : hence
Ac
which agrees with first formula established in this article.
262. An indefinite integral may also be diflferentiated with
respect to a variable contained in the function under the sign
of integration which is independent of the variable to which
the integration refers.
Let the integral be u =.Jf(^x, t)dx, t being independent of
X : then, without impairing the generality of this integral, we
may write
uz=z r /{x,i)dx + xp{t);
«/ a
\p{t) being an arbitrary function of t. Differentiating with
respect to /, t not depending on a, we have (Art. 260)
but, since ^p^{t) is a constant with respect to a?, it may be
included in the constant of the integral / *^\^ ^ dx; and
hence the last equation may be written
du_rd/{x,t).
di'-J ~dl~'^'''
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INTEGRATION UNDER THE SIGN f.
449
and we have only to differentiate the function under the sign J
with respect to U
263. Integration under the Sign of Integra^
/b
f(x^y)dx as the differ-
ential co-eflBicient of y, and integrating, we have
JdyJ ^f{x,y)dx
for our result ; and it is proposed to prove that this result is
the same in whichever order with respect to x and y the inte-
grations are performed ; that is, we shall have
i^^i a ^^^' y)^=^ ^dx^ fix, y)dy.
J{x,y)dx.
Integrating the two members of this equation with respect to
y, we get
J dxjf(x, y) dy = J <^y j /{^, y)dx;
and, if the limiting values of y are c and c?, we shall have
f ^dxj'f{x,y)dy
=J dy J f{x,y)dx.
The figure gives the geometri-
cal interpretation of this formula.
Either member represents the vol- ^
ume AG' included between the '^ ""
plane (x,y), the surface A'B'C'D' having z =zf{x,y) for its
57
z
I
â–º'
jL
A
^ "J^
c
^
/ /
/
3
(S
A
D
/
Digitized by VjOOQIC
450 INTEGRAL CALCULUS.
equation, and the planes whose equations are 05= a, a? = 6,
y=ic, y = d.
Example 1. Find the form of the function qp(x) such that
tlio area included between the curve y=:q)(^x)j the axis of Xj
and the ordinates y = 0, y =: qp(a), shall bear a constant ratio,
rij to the rectangle contained by the latter ordinate and the
corresponding abscissa.
By the conditions, we must have
Jo u
and, since this is to hold for all values of a, we may diflFeren-
tiate with respect to a : hence
^ ^ ^ n n
cp^(a) _ n — 1 ^
g)(a) a
and by integration
Zgp(a) =: (n — 1) Za -|- (7.
Passing from logarithms to numbers,
g)(a)= Ca^-^r r . (p{x) = Cx''-'^]
and the equation of the curve is y =. Cx*^~^,
Ex. 2. Determine such a form forqp(aj) that the integral
u= f -y/ — - shall be independent of a.
J \/{a — x)
Put x = az; then, since the limits a; = 0, a; = a, correspond
to 2 = 0, a == 1,
f»'* (p(x)dx f^ \/aq)(az)dz
^^ ^ r (p{X)ax _ /•
V(«-^)""^o V(i-»)
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INTEGRATION UNDER THE SIGN f.
451
By condition, u is to bo independent of a; therefore the dif-
ferential co-efficiont of u, with respect to a, must be zero.
But
^1
da
(^(az)
2^ a
+ zs/acp\az)
V(i-2)
_ /.<'f(xH-2xg/(a;) ,
and, since this last integral is to be zero for all values of a,
we must have
,(.) + 2x,'(x)=0:...^)=-^
2a!'
Therefore
l(f{x) = — -Ix -{- C,
or • 9'(^) = :^-
Let AOB bo a cycloid, with its vertex downwards ; and let it
be referred to the axis
Ox, and the tangent
through its vertex, as rr
co-ordinate axes. Pr
Then, denoting the
angle DCP by 0, we V K ^h"
have for the co-ordinates 0F:= y, OQ = x of the point P,
x = FP = HL = a-\-a cos.^,
y=iOF:=AR-^AE—AR - -AD + ED
^aTi-^ad -\-a sin. d.
Put = 7Z — g), then these values of x and ^/ become
x^=za — acos.9) (1), y z= a:/) + «8in.gj (2),
From (1) we find
, a — a? . 1 I '
g)z=:cos. ^ , sm.g)=:- 2aa; — ic^ ;
a a \
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452 INTEGRAL CALCULUS.
and thus (2) becomes
y = a cos -^ [- 2ax — aJ^
which is the equation of the cycloid. By differentiation, we
get
and, by integration, s =z \/Sax. We therefore conclude that
cp{x), in Ex. 2, is the expression for the arc of a cycloid esti-
mated from the vertex.
This example is the solution of the problem in mechanics
for finding the curve down which bodies, starting from dif-
ferent points, will fall in equal times.
264. The JSulerian Integral of the First Species
is an integral of the form
f xp-\i — xy-^dx,
J
in which p and q are positive numbers. This is denoted by
Bip, q).
The Eulerian Integral of the Second Species is
of the form
and is denoted by r{n).
The first species miy be put under the two forms
by making: x = ^^ — for the first form, and x := sin.**? for the
â– ^ ^ 1 + y
second.
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EULERIAN INTEGRALS. 453
The integral of the first species is a symmetrical function
of p and q; for, making a: = 1 — y, we have
B{p,q) = B{q,p).
26S. Integrating by parts, we have
dx
= - ^^^^V|>'- (1 -X )-ui - -)
_ _ 'g^(l-^)' ^.P/'a.i.-i(i_a;)«-i(^x - gfa;''(l-a;)'-'<fa;.
Therefore, taking 1 and for the limits, we have
B{p-\-\,q) = ^B{p,q)-^B{p-\-l,q):
.'. Bip+l,q) = ^Bip,q).
In like manner,
Bip,q+l)=^Bip,q).
In the integral of the first species, therefore, each of the
exponents p and q may be diminished by unity.
26G» In the Eulerian integral of the second species
/e"' x"^-^ dx,
n must be positive, otherwise the integral would be infinite.
For if n be negative, and equal to — p, we should have
J ^^'^
and it is plain, that, when oj = oo , the diff'erential co-efficient is
zero, and therefore the integral is zero ; and, when a? = 0, the
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454 INTEGRAL CALCULUS.
differential coeflScient is infinite; and therefore the integral
i3 infinite.
The integral /"(n + l) may bo made to depend on r{n).
For, integrating by parts, we have
But e'^x"^ reduces to zero both when a? = and when a? = oo
(Ex. 3, Art. 103) : therefore
e-'x'^dx = 711 e^^'x^^^dx,
«/
or r{n + 1) = nr{n).
In like manner,
r{n) = (n - l)r(n - 1), r{n - 1) = (ai - 2)r(n - 2) ;
and, if ti is entire, we shall have, finally,
r{2) = r(i), r(i) = T e— d^ = i.
Therefore, when n is an entire and positive number, we shall
have
r(?i) = 1.2.3. ..(n- 1);
and, if n is a fraction greater than 1, then the formula
r(n) = (n— l)r(n — 1)
enables us to reduce the integral r(n) to that of ./^(f*), f* de-
noting a number less than 1. Hence, to compute the value
r{n)y it is sufficient to know the values of this function for
values of n between and 1.
267. By putting e""-^ = y, the integral r(n) may bo made
to take another form. Thus, from e"""^ = y, we get
y y
... jy''^-^dx=^Jl(liydy =
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or
EULERIAN INTEGRALS, 455
2€8. Jtelations between the two JEulerian Inte"
grals. Assume the double integral
and integrate with respect to x : it thus becomes
Integrating the same double integral with respect to y, it be-
comes
, %. =-^(^)/, e-^x^-'dx = r{p)r{q):