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502 Chapter7
IntegralsofFunctionsofSeveralVariables
WecannowcompletetheproofofTheorem7.3.8. Lemmas7.3.13and7.3.14imply
(7.3.28)iff isnonnegativeonS.Nowsupposethat
mDmin
˚
f.X/
ˇ
ˇ
X2G.S/
<0:
ThenfmisnonnegativeonG.S/,so(7.3.28)withfreplacedbyf mimpliesthat
Z
G.S/
.f.X/m/dXD
Z
S
.f.G.Y/m/jJG.Y/jdY:
(7.3.47)
However,settingf D1in(7.3.28)yields
Z
G.S/
dXD
Z
S
jJG.Y/jdY;
so(7.3.47)implies(7.3.28).
TheassumptionsofTheorem7.3.8aretoostringentformanyapplications.Forexample,
tofindtheareaofthedisc
˚
.x;y/
ˇ
ˇ
x
2
Cy
2
1
;
itisconvenienttousepolarcoordinatesandregardthecircleasG.S/,where
G.r;/D
rcos
rsin
(7.3.48)
andSisthecompactset
SD
˚
.r;/
ˇ
ˇ
0r1;02
(7.3.49)
(Figure7.3.3).
S
X = G(r, θ)
1
θ
y
r
x
x2 + y2 = 1
G(S)
Figure7.3.3
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Section7.3
ChangeofVariablesinMultipleIntegrals
503
Since
G
0
.r;/D
cos rsin
sin
rcos
;
itfollowsthatJG.r;/ D r. Therefore,formallyapplyingTheorem7.3.8withf 1
yields
AD
Z
G.S/
dXD
Z
S
rd.r;/D
Z
1
0
rdr
Z
2
0
dD:
Althoughthisisafamiliarresult,Theorem7.3.8doesnotreallyapplyhere,sinceG.r;0/D
G.r;2/,0r1,soGisnotone-to-oneonS,andthereforenotregularonS.
ThenexttheoremshowsthattheassumptionsofTheorem7.3.8canberelaxedsoasto
includethisexample.
Theorem7.3.15
SupposethatG W E
n
! R
n
iscontinuouslydifferentiableona
boundedopensetN containingthecompactJordanmeasurablesetS;andregular r on
S
0
:SupposealsothatG.S/isJordanmeasurable;f iscontinuousonG.S/;andG.C/is
JordanmeasurableforeverycubeC N.Then
Z
G.S/
f.X/dXD
Z
S
f.G.Y//jJG.Y/jdY:
(7.3.50)
Proof
Sincef iscontinuousonG.S/and.jJGj/fıGiscontinuousonS,theintegrals
in(7.3.50)bothexist,byCorollary7.3.2.Nowlet
Ddist.@S;N
c
/
(Exercise5.1.25),and
P D
˚
Y
ˇ
ˇ
dist.Y;@S/
2
:
ThenPisacompactsubsetofN(Exercise5.1.26)and@SP0(Figure7.3.4).
SinceSisJordanmeasurable,V.@S/D0,byTheorem7.3.1.Therefore,if>0,we
canchoosecubesC
1
,C
2
,...,C
k
inP
0
suchthat
@S
[k
jD1
C
0
j
(7.3.51)
and
Xk
jD1
V.C
j
/<
(7.3.52)
NowletS
1
betheclosureofthesetofpointsinSthatarenotinanyofthecubesC
1
,
C
2
,...,C
k
;thus,
S
1
D
S\
[
k
jD1
C
j
c
:
504 Chapter7
IntegralsofFunctionsofSeveralVariables
Becauseof(7.3.51),S
1
\@S D;,soS
1
isacompactJordanmeasurablesubsetofS
0
.
Therefore,GisregularonS
1
,andf iscontinuousonG.S
1
/. Consequently,ifQisas
definedin(7.3.37),thenQ.S
1
/D0byTheorem7.3.8.
N = open set bounded by outer curve
S = closed set bounded by inner curve
∂S
D
ρ
Figure7.3.4
Now
Q.S/DQ.S
1
/CQ.S\S
c
1
/DQ.S\S
c
1
/
(7.3.53)
(Exercise7.3.11)and
jQ.S\S
c
1
/j
ˇ
ˇ
ˇ
ˇ
ˇ
Z
G.S\S
c
1
/
f.X/dX
ˇ
ˇ
ˇ
ˇ
ˇ
C
ˇ
ˇ
ˇ
ˇ
ˇ
Z
S\S
c
1
f.G.Y//jJG.Y/jdY
ˇ
ˇ
ˇ
ˇ
ˇ
:
But
ˇ
ˇ
ˇ
ˇ
ˇ
Z
S\S
c
1
f.G.Y//jJG.Y/jdY
ˇ
ˇ
ˇ
ˇ
ˇ
M
1
M
2
V.S\S
c
1
/;
(7.3.54)
whereM
1
andM
2
areas definedin(7.3.38)and(7.3.39). SinceS\S
c
1
 [
k
jD1
C
j
,
(7.3.52)impliesthatV.S\S
k
1
/<;therefore,
ˇ
ˇ
ˇ
ˇ
ˇ
Z
S\S
c
1
f.G.Y//jJG.Y/jdY
ˇ
ˇ
ˇ
ˇ
ˇ
M
1
M
2
;
(7.3.55)
from(7.3.54).Also
ˇ
ˇ
ˇ
ˇ
ˇ
Z
G.S\S
c
1
/
f.X/dX
ˇ
ˇ
ˇ
ˇ
ˇ
M
2
V.G.S\S
c
1
//M
2
Xk
jD1
V.G.C
j
//:
(7.3.56)
Section7.3
ChangeofVariablesinMultipleIntegrals
505
Bytheargumentthatledto(7.3.30)withH DGandCDC
j
,
V.G.C
j
//
max
˚
kG
0
.Y/k
1
ˇ
ˇ
Y2C
j

n
V.C
j
/;
so(7.3.56)canberewrittenas
ˇ
ˇ
ˇ
ˇ
ˇ
Z
G.S\S
c
1
/
f.X/dX
ˇ
ˇ
ˇ
ˇ
ˇ
M
2
max
˚
kG
0
.Y/k
1
ˇ
ˇ
Y2P

n
;
becauseof(7.3.52). Sincecanbemadearbitrarilysmall,thisand(7.3.55)implythat
Q.S\S
c
1
/D0.NowQ.S/D0,from(7.3.53).
Thetransformationtopolarcoordinatestocomputetheareaofthediscisnowjusti-
fied,sinceGandSasdefinedby(7.3.48)and(7.3.49)satisfytheassumptionsofTheo-
rem7.3.15.
PolarCoordinates
IfGisthetransformationfrompolartorectanglecoordinates
x
y
DG.r;/D
rcos
rsin
;
(7.3.57)
thenJG.r;/Drand(7.3.50)becomes
Z
G.S/
f.x;y/d.x;y/D
Z
S
f.rcos;rsin/rd.r;/
ifweassume,asisconventional,thatSisintheclosedrighthalfofther-plane. This
transformationisespeciallyusefulwhentheboundariesofScanbeexpressedconveniently
interms ofpolarcoordinates,asintheexampleprecedingTheorem 7.3.15. Twomore
examplesfollow.
Example7.3.2
Evaluate
ID
Z
T
.x
2
Cy/d.x;y/;
whereTistheannulus
TD
˚
.x;y/
ˇ
ˇ
1x
2
Cy
2
4
(Figure7.3.5
(b)
).
Solution
WewriteT DG.S/,withGasin(7.3.57)and
SD
˚
.r;/
ˇ
ˇ
1r2;02
506 Chapter7
IntegralsofFunctionsofSeveralVariables
(Figure7.3.5
(a)
).Theorem7.3.15impliesthat
ID
Z
S
.r
2
cos
2
Crsin/rd.r;/;
whichweevaluateasaniteratedintegral:
ID
Z
2
1
r
2
dr
Z
2
0
.rcos
2
Csin/d
D
Z
2
1
r
2
dr
Z
2
0
r
2
C
r
2
cos2Csin
d
since cos
2
D
1
2
.1Ccos2/
D
Z
2
1
r
2
r
2
C
r
4
sin2cos
ˇ
ˇ
ˇ
ˇ
2
D0
drD
Z
2
1
r
3
drD
r
4
4
ˇ
ˇ
ˇ
ˇ
2
1
D
15
4
:
T
S
y
r
x
(a)
(b)
θ
2
1
Figure7.3.5
Example7.3.3
Evaluate
I D
Z
T
yd.x;y/;
whereTistheregioninthexy-planeboundedbythecurvewhosepointshavepolarcoor-
dinatessatisfying
rD1cos; 0
(Figure7.3.6
(b)
).
Solution
WewriteT D D G.S/, , withG as in(7.3.57)andS theshadedregionin
Figure7.3.6
(a)
.From(7.3.50),
I D
Z
S
.rsin/rd.r;/;
Section7.3
ChangeofVariablesinMultipleIntegrals
507
whichweevaluateasaniteratedintegral:
I D
Z
0
sind
Z
1cos
0
r
2
drD
1
3
Z
0
.1cos/
3
sind
D
1
12
.1cos/
4
ˇ
ˇ
ˇ
ˇ
0
D
4
3
:
T
S
r
y
x
θ
π
(b)
(a)
Figure7.3.6
SphericalCoordinates
IfGisthetransformationfromsphericaltorectangularcoordinates,
2
4
x
y
´
3
5
DG.r;;/D
2
4
rcoscos
rsincos
rsin
3
5
;
(7.3.58)
then
G
0
.r;;/D
2
4
coscos rsincos rcossin
sincos
rcoscos rsinsin
sin
0
rcos
3
5
andJG.r;;/Dr2cos,so(7.3.50)becomes
Z
G.S/
f.x;y;´/d.x;y;´/
D
Z
S
f.rcoscos;rsincos;rsin/r
2
cosd.r;;/
(7.3.59)
ifwemaketheconventionalassumptionthatjj=2andr0.
508 Chapter7
IntegralsofFunctionsofSeveralVariables
Example7.3.4
Leta>0.Findthevolumeof
T D
˚
.x;y;´/
ˇ
ˇ
x
2
Cy
2
2
a
2
;x0;y0;´0
;
whichisoneeighthofasphere(Figure7.3.7
(b)
).
(a)
(b)
y
z
φ
θ
r
x
2
π
2
π
a
a
a
a
Figure7.3.7
Solution
WewriteT DG.S/withGasin(7.3.58)and
SD
˚
.r;;/
ˇ
ˇ
0ra;0=2;0=2
Section7.3
ChangeofVariablesinMultipleIntegrals
509
(Figure7.3.7
(a)
),andletf 1in(7.3.59).Theorem7.3.15impliesthat
V.T/D
Z
G.S/
dXD
Z
S
r
2
cosd.r;;/
D
Z
a
0
r
2
dr
Z
=2
0
d
Z
=2
0
cosdD
a
3
3
2
(7.3.1)D
a
3
6
:
Example7.3.5
Evaluatetheiteratedintegral
I D
Z
a
0
xdx
Z
p
a
2
x
2
0
dy
Z
p
a
2
x
2
y
2
0
´d´ .a>0/:
Solution
WefirstrewriteIasamultipleintegral
I D
Z
G.S/
x´d.x;y;´/
whereGandSareasinExample7.3.4.FromTheorem7.3.15,
ID
Z
S
.rcoscos/.rsin/.r
2
cos/d.r;;/
D
Z
a
0
r
4
dr
Z
=2
0
cosd
Z
=2
0
cos
2
sindD
a
5
5
(7.3.1)
1
3
D
a
5
15
:
OtherExamples
WenowconsiderotherapplicationsofTheorem7.3.15.
Example7.3.6
Evaluate
ID
Z
T
.xC4y/d.x;y/;
whereTistheparallelogramboundedbythelines
xCyD1; xCyD2; x2yD0; and x2yD3
(Figure7.3.8
(b)
).
Solution
Wedefinenewvariablesuandvby
u
v
DF.x;y/D
xCy
x2y
:
510 Chapter7
IntegralsofFunctionsofSeveralVariables
S
v
u
2
3
1
(a)
x
y
= F−1(u,v)
T
y
x
(b)
x − 2y = 0
x − 2y = 3
x + y = 2
x + y = 1
Figure7.3.8
Then
x
y
DF
1
.u;v/D
2
6
4
2uCv
3
uv
3
3
7
5
;
JF
1
.u;v/D
ˇ
ˇ
ˇ
ˇ
ˇ
2
3
1
3
1
3
1
3
ˇ
ˇ
ˇ
ˇ
ˇ
D
1
3
;
andT DF
1
.S/,where
SD
˚
.u;v/
ˇ
ˇ
1u2;0v3
Section7.3
ChangeofVariablesinMultipleIntegrals
511
(Figure7.3.8
(a)
).ApplyingTheorem7.3.15withGDF
1
yields
I D
Z
S
2uCv
3
C
4u4v
3

1
3
d.u;v/D
1
3
Z
S
.2uv/d.u;v/
D
1
3
Z
3
0
dv
Z
2
1
.2uv/duD
1
3
Z
3
0
.u
2
uv/
ˇ
ˇ
ˇ
ˇ
2
uD1
dv
D
1
3
Z
3
0
.3v/dvD
1
3
3v
v
2
2
ˇ
ˇ
ˇ
ˇ
3
0
D
3
2
:
Example7.3.7
Evaluate
ID
Z
T
e
.x
2
y
2
/
2
e
4x
2
y
2
.x
2
Cy
2
/d.x;y/;
whereT istheannulusT D
˚
.x;y/
ˇ
ˇ
a
2
x
2
Cy
2
b
2
witha >0andb >0(Fig-
ure7.3.9
(a)
).
y
x
y
x
a
b
T
a
b
T
1
T
2
T
3
T
4
(a)
(b)
Figure7.3.9
Solution
Theformsofthearguments oftheexponentialfunctionssuggestthatwe
introducenewvariablesuandvdefinedby
u
v
DF.x;y/D
x
2
y
2
2xy
andapplyTheorem7.3.15toGDF1.However,Fisnotone-to-oneonTandtherefore
hasnoinverseonT0(Example6.3.4).Toremovethisdifficulty,weregardT astheunion
ofthequarter-annuliT
1
,T
2
,T
3
,andT
4
inthefourquadrants(Figure7.3.9)
(b)
),andlet
I
j
D
Z
T
j
e
.x
2
y
2
/
2
e
4x
2
y
2
.x
2
Cy
2
/d.x;y/:
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