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472 Chapter7
IntegralsofFunctionsofSeveralVariables
FromTheorem7.1.22,theassumptionsofTheorem7.2.6aresatisfiediff iscontinuous
onSanduandvarecontinuouslydifferentiableonŒc;d.
InterchangingxandyinTheorem7.2.6showsthatiff isintegrableon
SD
˚
.x;y/
ˇ
ˇ
u.x/yv.x/;axb
(7.2.21)
(Figure7.2.2)and
Z
v.x/
u.x/
f.x;y/dy
existsforaxb,then
Z
S
f.x;y/d.x;y/D
Z
b
a
dx
Z
v.x/
u.x/
f.x;y/dy:
(7.2.22)
y
x
a
b
y = v(x)
y = u(x)
S
Figure7.2.2
Example7.2.6
Supposethat
f.x;y/Dxy
andSistheregionboundedbythecurvesxDy
2
andxDy(Figure7.2.3).SinceScan
berepresentedintheform(7.2.17)as
SD
˚
.x;y/
ˇ
ˇ
y
2
xy;0y1
;
(7.2.20)yields
Z
S
xyd.x;y/D
Z
1
0
dy
Z
y
y2
xydx;
which,incidentally,canbewrittenas
Z
S
xyd.x;y/D
Z
1
0
ydy
Z
y
y2
xdx;
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Section7.2
IteratedIntegralsandMultipleIntegrals
473
sinceyisindependentofx.Evaluatingtheiteratedintegralyields
Z
S
xyd.x;y/D
Z
1
0
x
2
2
ˇ
ˇ
ˇ
ˇ
y
y
2
!
ydyD
1
2
Z
1
0
.y
3
y
5
/dy
D
1
2
y
4
4
y
6
6
ˇ
ˇ
ˇ
ˇ
1
0
D
1
24
:
y
x
x = y2
x = y
(1, 1)
S
Figure7.2.3
InthiscasewecanalsorepresentSintheform(7.2.21)as
SD
˚
.x;y/
ˇ
ˇ
xy
p
x;0x1
I
hence,from(7.2.22),
Z
S
xyd.x;y/D
Z
1
0
xdx
p
x
x
ydyD
Z
1
0
y
2
2
ˇ
ˇ
ˇ
ˇ
p
x
yDx
!
xdx
D
1
2
Z
1
0
.x
2
x
3
/dxD
1
2
x
3
3
x
4
4
ˇ
ˇ
ˇ
ˇ
1
0
D
1
24
:
Example7.2.7
Toevaluate
Z
S
.xCy/d.x;y/;
where
SD
˚
.x;y/
ˇ
ˇ
1x1;0y1Cjxj
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474 Chapter7
IntegralsofFunctionsofSeveralVariables
(seeExample7.1.11andFigure7.2.4),
S
y
x
y = 1 − x
y = 1 + x
1
−1
Figure7.2.4
weinvokeCorollary7.1.31andwrite
Z
S
.xCy/d.x;y/D
Z
S
1
.xCy/d.x;y/C
Z
S
2
.xCy/d.x;y/;
where
S
1
D
˚
.x;y/
ˇ
ˇ
0x1;0y1Cx
and
S
2
D
˚
.x;y/
ˇ
ˇ
1x0;0y1x
(Figure7.2.5).
FromTheorem7.2.6,
Z
S
1
.xCy/d.x;y/D
Z
1
0
dx
Z
1Cx
0
.xCy/dyD
Z
1
0
"
.xCy/2
2
ˇ
ˇ
ˇ
ˇ
1Cx
yD0
#
dx
D
1
2
Z
1
0
.2xC1/
2
x
2
dx
D
1
2
.2xC1/3
6
x3
3
ˇ
ˇ
ˇ
ˇ
1
0
D2
and
Z
S
2
.xCy/d.x;y/D
Z
0
1
dx
Z
1x
0
.xCy/dyD
Z
0
1
"
.xCy/
2
2
ˇ
ˇ
ˇ
ˇ
1x
yD0
#
dx
D
1
2
Z
0
1
.1x
2
/dxD
1
2
x
x
3
3
ˇ
ˇ
ˇ
ˇ
0
1
D
1
3
:
Therefore,
Z
S
.xCy/d.x;y/D2C
1
3
D
7
3
:
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Section7.2
IteratedIntegralsandMultipleIntegrals
475
y
y
x
x
y = 1 − x
y = 1 + x
S
1
S
2
Figure7.2.5
Example7.2.8
TofindtheareaAoftheregionboundedbythecurves
yDx
2
C1 and yD9x
2
(Figure7.2.6),weevaluate
AD
Z
S
d.x;y/;
where
S D
˚
.x;y/
ˇ
ˇ
x
2
C1y9x
2
;2x2
:
AccordingtoTheorem7.2.6,
AD
Z
2
2
dx
Z
9x
2
x
2
C1
dyD
Z
2
2
.9x
2
/.x
2
C1/
dx
D
Z
2
2
.82x
2
/dxD
8x
2x
3
3
ˇ
ˇ
ˇ
ˇ
2
2
D
64
3
:
y
x
y = x2 + 1
y = 9 − x2
(2, 5)
(−2, 5)
S
Figure7.2.6
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476 Chapter7
IntegralsofFunctionsofSeveralVariables
Theorem7.2.6hasananalogforn>2.SupposethatfisintegrableonasetSofpoints
XD.x
1
;x
2
;:::;x
n
/satisfyingtheinequalities
u
j
.x
jC1
;:::;x
n
/x
j
v
j
.x
jC1
;:::;x
n
/; 1j j n1;
and
a
n
x
n
b
n
:
Then,underappropriateadditionalassumptions,itcanbeshownbyanargumentanalogous
totheonethatledtoTheorem7.2.6that
Z
S
f.X/dXD
Z
b
n
a
n
dx
n
Z
v
n
.x
n
/
u
n
.x
n
/
dx
n1

Z
v
2
.x
3
;:::;x
n
/
u
2
.x
3
;:::;x
n
/
dx
2
Z
v
1
.x
2
;:::;x
n
/
u
1
.x
2
;:::;x
n
/
f.X/dx
1
:
Theseadditionalassumptionsaretedioustostateforgeneraln. Thefollowingtheorem
containsacompletestatementfornD3.
Theorem7.2.7
Supposethatf isintegrableon
SD
˚
.x;y;´/
ˇ
ˇ
u
1
.y;´/xv
1
.y;´/; u
2
.´/yv
2
.´/; c´d
;
andlet
S.´/D
˚
.x;y/
ˇ
ˇ
u
1
.y;´/xv
1
.y;´/;u
2
.´/yv
2
.´/
foreach´inŒc;d:Then
Z
S
f.x;y;´/d.x;y;´/D
Z
d
c
Z
v
2
.´/
u
2
.´/
dy
Z
v
1
.y;´/
u
1
.y;´/
f.x;y;´/dx;
providedthat
Z
v
1
.y;´/
u
1
.y;´/
f.x;y;´/dx
existsforall.y;´/suchthat
c´d and u
2
.´/yv
2
.´/;
and
Z
S.´/
f.x;y;´/d.x;y/
existsforall´inŒc;d:
Example7.2.9
Supposethatf iscontinuousontheregionS S inR
3
boundedbythe
coordinateplanesandtheplane
xCyC2´D2
(Figure7.2.7);thus,
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Section7.2
IteratedIntegralsandMultipleIntegrals
477
y
x
z
x + y + 2z = 1
Figure7.2.7
SD
˚
.x;y;´/
ˇ
ˇ
0x2y2´; 0y22´;0´1
:
FromTheorem7.2.7,
Z
S
f.x;y;´/d.x;y;´/D
Z
1
0
Z
22´
0
dy
Z
2y2´
0
f.x;y;´/dx:
Therearefiveotheriteratedintegralsthatequalthemultipleintegral.Weleaveittoyou
toverifythat
Z
S
f.x;y;´/d.x;y;´/D
Z
2
0
dy
Z
1y=2
0
Z
2y2´
0
f.x;y;´/dx
D
Z
1
0
Z
22´
0
dx
Z
2x2´
0
f.x;y;´/dy
D
Z
2
0
dx
Z
1x=2
0
Z
2x2´
0
f.x;y;´/dy
D
Z
2
0
dx
Z
2x
0
dy
Z
1x=2y=2
0
f.x;y;´/d´
D
Z
2
0
dy
Z
2y
0
dx
Z
1x=2y=2
0
f.x;y;´/d´
(Exercise7.2.15).
Thusfarwehaveviewedtheiteratedintegralasatoolforevaluatingmultipleintegrals.
Insomeproblemstheiteratedintegralisitselftheobjectofinterest. Inthiscasearesult
478 Chapter7
IntegralsofFunctionsofSeveralVariables
likeTheorem7.2.6canbeusedtoevaluatetheiteratedintegral.Theprocedureisasfollows.
(a)
Expressthegiveniteratedintegralasamultipleintegral,andchecktoseethatthe
multipleintegralexists.
(b)
Lookforanotheriteratedintegralthatequalsthemultipleintegralandiseasierto
evaluatethan thegiven one. Thetwoiteratedintegralsmust t be equal, byTheo-
rem7.2.6.
Thisprocedureiscalledchangingtheorderofintegrationofaniteratedintegral.
Example7.2.10
Theiteratedintegral
I D
Z
1
0
dy
Z
y
0
e
.x1/
2
dx
ishardtoevaluatebecausee
.x1/
2
hasnoelementaryantiderivative. Thesetofpoints
.x;y/thatenterintotheintegration,whichwecalltheregionofintegration,is
SD
˚
.x;y/
ˇ
ˇ
0xy;0y1
(Figure7.2.8).
y
x
y = x
1
1
Figure7.2.8
Therefore,
ID
Z
S
e
.x1/
2
d.x;y/;
(7.2.23)
andthismultipleintegralexistsbecauseitsintegrandiscontinuous. SinceScanalsobe
writtenas
SD
˚
.x;y/
ˇ
ˇ
xy1;0x1
;
Section7.2
IteratedIntegralsandMultipleIntegrals
479
Theorem7.2.6impliesthat
Z
S
e
.x1/
2
d.x;y/D
Z
1
0
e
.x1/
2
dx
Z
1
x
dyD
Z
1
0
.x1/e
.x1/
2
dx
D
1
2
e
.x1/
2
ˇ
ˇ
ˇ
ˇ
1
0
D
1
2
.1e
1
/:
Thisand(7.2.23)implythat
I D
1
2
.1e
1
/:
Example7.2.11
Supposethatf iscontinuousonŒa;1/andysatisfiesthedifferen-
tialequation
y
00
.x/Df.x/; x>a;
(7.2.24)
withinitialconditions
y.a/Dy
0
.a/D0:
Integrating(7.2.24)yields
y
0
.x/D
Z
x
a
f.t/dt;
sincey
0
.a/D0.Integratingthisyields
y.x/D
Z
x
a
ds
Z
s
a
f.t/dt;
sincey.a/D0.Thiscanbereducedtoasingleintegralasfollows.Sincethefunction
g.s;t/Df.t/
iscontinuousforall.s;t/suchthatta,gisintegrableon
SD
˚
.s;t/
ˇ
ˇ
ats;asx
(Figure7.2.9),andTheorem7.2.6impliesthat
Z
S
f.t/d.s;t/D
Z
x
a
ds
Z
s
a
f.t/dtDy.x/:
(7.2.25)
However,Scanalsobedescribedas
SD
˚
.s;t/
ˇ
ˇ
tsx; atx
soTheorem7.2.6impliesthat
Z
S
f.t/d.s;t/D
Z
x
a
f.t/dt
Z
x
t
dsD
Z
x
a
.xt/f.t/dt:
Comparingthiswith(7.2.25)yields
y.x/D
Z
x
a
.xt/f.t/dt:
480 Chapter7
IntegralsofFunctionsofSeveralVariables
t
x
s
a
a
s = t
S
Figure7.2.9
7.2Exercises
1.
Evaluate
(a)
Z
2
0
dy
Z
1
1
.xC3y/dx
(b)
Z
2
1
dx
Z
1
0
.x
3
Cy
4
/dy
(c)
Z
2
=2
xdx
Z
2
1
sinxydy
(d)
Z
log2
0
ydy
Z
1
0
xe
x
2
y
dx
2.
LetI
j
DŒa
j
;b
j
,1j 3,andsupposethatfisintegrableonRDI
1
I
2
I
3
.
Prove:
(a)
Iftheintegral
G.y;´/D
Z
b
1
a
1
f.x;y;´/dx
existsfor.y;´/2I
2
I
3
,thenGisintegrableonI
2
I
3
and
Z
R
f.x;y;´/d.x;y;´/D
Z
I
2
I
3
G.y;´/d.y;´/:
(b)
Iftheintegral
H.´/D
Z
I
1
I
2
f.x;y;´/d.x;y/
Section7.2
IteratedIntegralsandMultipleIntegrals
481
existsfor´2I
3
,thenHisintegrableonI
3
and
Z
R
f.x;y;´/d.x;y;´/D
Z
b
3
a
3
H.´/d´:
H
INT
:Forbothparts;seetheproofofTheorem7.2.1:
3.
Prove:Iff iscontinuousonŒa;bŒc;d,thenthefunction
F.y/D
Z
b
a
f.x;y/dx
iscontinuousonŒc;d.H
INT
:UseTheorem5.2.14:
4.
Supposethat
f.x
0
;y
0
/f.x;y/ if axx
0
b;cyy
0
d:
ShowthatfsatisfiesthehypothesesofTheorem7.2.1onRDŒa;bŒc;d.H
INT
:
SeetheproofofTheorem3.2.9:
5.
Evaluatebymeansofiteratedintegrals:
(a)
Z
R
.xyC1/d.x;y/; RDŒ0;1Œ1;2
(b)
Z
R
.2xC3y/d.x;y/; RDŒ1;3Œ1;2
(c)
Z
R
xy
p
x2Cy2
d.x;y/; RDŒ0;1Œ0;1
(d)
R
R
xcosxycos2xd.x;y/; RDŒ0;
1
4
Œ0;2
6.
LetAbethesetofpointsoftheform.2
m
p;2
m
q/,wherepandqareoddintegers
andmisanonnegativeinteger.Let
f.x;y/D
(
1; .x;y/62A;
0; .x;y/2A:
Showthatf isnotintegrableonanyrectangleRDŒa;bŒc;d,but
Z
b
a
dx
Z
d
c
f.x;y/dyD
Z
d
c
dy
Z
b
a
f.x;y/dxD.ba/.d c/:
.A/
H
INT
:For(A);useTheorem3.5.6andExercise3.5.6:
7.
Let
f.x;y/D
2xy
ifyisrational;
y
ifyisirrational;
andRDŒ0;1Œ0;1(Example7.2.3).
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