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482 Chapter7
IntegralsofFunctionsofSeveralVariables
(a)
Calculate
R
R
f.x;y/d.x;y/and
R
R
f.x;y/d.x;y/,andshowthatf isnot
integrableonR.
(b)
Calculate
R
1
0
R
1
0
f.x;y/dy
dxand
R
1
0
R
1
0
f.x;y/dy
dx.
8.
LetRDŒ0;1Œ0;1Œ0;1,
e
RDŒ0;1Œ0;1,and
f.x;y;´/D
8
ˆ
ˆ
<
ˆ
ˆ
:
2xyC2x´
ifyand´arerational;
yC2x´
ifyisirrationaland´isrational;
2xyC´
ifyisrationaland´isirrational;
yC´
ifyand´areirrational:
Calculate
(a)
Z
R
f.x;y;´/d.x;y;´/and
Z
R
f.x;y;´/d.x;y;´/
(b)
Z
e
R
f.x;y;´/d.x;y/and
Z
e
R
f.x;y;´/d.x;y/
(c)
Z
1
0
dy
Z
1
0
f.x;y;´/dxand
Z
1
0
Z
1
0
dy
Z
1
0
f.x;y;´/dx.
9.
Supposethatf isboundedonRDŒa;bŒc;d.Prove:
(a)
Z
R
f.x;y/d.x;y/
Z
b
a
Z
d
c
f.x;y/dy
!
dx.H
INT
:UseExercise3.2.6
(a)
:
(b)
Z
R
f.x;y/d.x;y/
Z
b
a
Z
d
c
f.x;y/dy
!
dx.H
INT
:UseExercise3.2.6
(b)
:
10.
UseExercise7.2.9toprovethefollowinggeneralizationofTheorem7.2.1:Iff is
integrableonRDŒa;bŒc;d,then
Z
b
a
f.x;y/dy and
Z
d
c
f.x;y/dy
areintegrableonŒa;b,and
Z
b
a
Z
d
c
f.x;y/dy
!
dxD
Z
b
a
Z
d
c
f.x;y/dy
!
dxD
Z
R
f.x;y/d.x;y/:
11.
Evaluate
(a)
Z
R
.x2yC3´/d.x;y;´/; RDŒ2;0Œ2;5Œ3;2
(b)
Z
R
e
x
2
y
2
sinxsin´d.x;y;´/; RDŒ1;1Œ0;2Œ0;=2
(c)
Z
R
.xyC2x´Cy´/d.x;y;´/; RDŒ1;1Œ0;1Œ1;1
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Section7.2
IteratedIntegralsandMultipleIntegrals
483
(d)
Z
R
x
2
y
3
´e
xy
2
´
2
d.x;y;´/; RDŒ0;1Œ0;1Œ0;1
12.
Evaluate
(a)
Z
S
.2xCy
2
/d.x;y/; SD
˚
.x;y/
ˇ
ˇ
0x9y
2
;3y3
(b)
Z
S
2xyd.x;y/; SisboundedbyyDx
2
andxDy
2
(c)
Z
S
e
x
siny
y
d.x;y/; SD
˚
.x;y/
ˇ
ˇ
logyxlog2y;=2y
13.
Evaluate
R
S
.xCy/d.x;y/,whereSisboundedbyyD x
2
andyD 2x,using
iteratedintegralsofbothpossibletypes.
14.
Findtheareaofthesetboundedbythegivencurves.
(a)
yDx
2
C9,yDx
2
9,xD1,xD1
(b)
yDxC2,yD4x,xD0
(c)
xDy
2
4,xD4y
2
(d)
yDe2x,yD2x,xD3
15.
InExample7.2.9,verifythelastfiverepresentationsof
R
S
f.x;y;´/d.x;y;´/as
iteratedintegrals.
16.
LetS betheregioninR
3
boundedbythecoordinateplanesandtheplanexC
2yC3´D 1. . Letf becontinuousonS. . Setupsixiteratedintegralsthatequal
R
S
f.x;y;´/d.x;y;´/.
17.
Evaluate
(a)
Z
S
xd.x;y;´/; Sisboundedbythecoordinateplanesandtheplane
3xCyC´D2.
(b)
Z
S
ye
´
d.x;y;´/;SD
˚
.x;y;´/
ˇ
ˇ
0x1;0y
p
x;0´y2
(c)
Z
S
xy´d.x;y;´/;
SD
n
.x;y;´/
ˇ
ˇ
0y1;0x
p
1y2;0´
p
x2Cy2
o
(d)
Z
S
y´d.x;y;´/;SD
˚
.x;y;´/
ˇ
ˇ
´
2
x
p
´; 0y´; 0´1
18.
FindthevolumeofS.
(a)
Sisboundedbythesurfaces´Dx
2
Cy
2
and´D8x
2
y
2
.
(b)
S Df.x;y;´/j0´x
2
Cy
2
; .x;y;0/isinthetrianglewithvertices
.0;1;0/,.0;0;0/,and.1;0;0/}
(c)
SD
˚
.x;y;´/
ˇ
ˇ
0yx
2
;0x2;0´y
2
(d)
SD
˚
.x;y;´/
ˇ
ˇ
x0;y0;0´44x
2
4y
2
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484 Chapter7
IntegralsofFunctionsofSeveralVariables
19.
LetRDŒa
1
;b
2
Œa
2
;b
2
Œa
n
;b
n
.Evaluate
(a)
R
R
.x
1
Cx
2
CCx
n
/dX
(b)
R
R
.x2
1
Cx2
2
CCx2
n
/dX
(c)
R
R
x
1
x
2
;x
n
dX
20.
Assumingthatf iscontinuous,express
Z
1
1=2
dy
Z
p
1y
2
p
1y2
f.x;y/dx
asaniteratedintegralwiththeorderofintegrationreversed.
21.
Evaluate
R
S
.xCy/d.x;y/ofExample7.2.7bymeansofiteratedintegralsinwhich
thefirstintegrationiswithrespecttox.
22.
Evaluate
Z
1
0
xdx
Z
p
1x
2
0
dy
p
x2Cy2
:
23.
Supposethatf iscontinuousonŒa;1/,
y
.n/
.x/Df.x/; ta;
andy.a/Dy
0
.a/DDy
.n1/
.a/D0.
(a)
Integraterepeatedlytoshowthat
y.x/D
Z
x
a
dt
n
Z
t
n
a
dt
n1

Z
t
3
a
dt
2
Z
t
2
a
f.t
1
/dt
1
:
.A/
(b)
BysuccessivereversalsofordersofintegrationasinExample7.2.11,deduce
from(A)that
y.x/D
1
.n1/Š
Z
x
a
.xt/
n1
f.t/dt:
24.
LetT
DŒ0;Œ0;;>0.Bycalculating
I.a/D lim
!1
Z
T
e
xy
sinaxd.x;y/
intwodifferentways,showthat
Z
1
0
sinax
x
dxD
2
if a>0:
7.3CHANGEOFVARIABLESINMULTIPLEINTEGRALS
InSection3.3wesawthatachangeofvariablesmaysimplifytheevaluationofanordinary
integral.Wenowconsiderchangeofvariablesinmultipleintegrals.
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Section7.3
ChangeofVariablesinMultipleIntegrals
485
Priortoformulatingtheruleforchangeofvariables,wemustdealwithsomerather
involvedpreliminaryconsiderations.
JordanMeasurableSets
InSection wedefinedthecontentofasetStobe
V.S/D
Z
S
dX
(7.3.1)
iftheintegralexists.IfRisarectanglecontainingS,then(7.3.1)canberewrittenas
V.S/D
Z
R
S
.X/dX;
where 
S
isthecharacteristicfunctionofS,definedby
S
.X/D
1; X2S;
0; X62S:
FromExercise7.1.27,theexistenceandvalueofV.S/donotdependontheparticular
choiceoftheenclosingrectangleR. WesaythatSisJordanmeasurableifV.S/exists.
ThenV.S/istheJordancontentofS.
Weleaveittoyou(Exercise7.3.2)toshowthatShaszerocontentaccordingtoDefini-
tion7.1.14ifandonlyifShasJordancontentzero.
Theorem7.3.1
AboundedsetSisJordanmeasurableifandonlyiftheboundaryof
Shaszerocontent:
Proof
LetRbearectanglecontainingS. SupposethatV.@S/ D 0. Since 
S
is
boundedonRanddiscontinuousonlyon@S(Exercise2.2.9),Theorem7.1.19impliesthat
R
R
S
.X/dXexists. Fortheconverse,supposethat@Sdoesnothavezerocontentand
letP DfR
1
;R
2
;:::;R
k
gbeapartitionofR.Foreachj inf1;2;:::;kgtherearethree
possibilities:
1.R
j
S;then
min
˚
S
.X/
ˇ
ˇ
X2R
j
Dmax
˚
S
.X/
ˇ
ˇ
X2R
j
D1:
2.R
j
\S¤;andR
j
\S
c
¤;;then
min
˚
S
.X/
ˇ
ˇ
X2R
j
D0 and max
˚
S
.X/
ˇ
ˇ
X2R
j
D1:
3.R
j
S
c
;then
min
˚
S
.X/
ˇ
ˇ
X2R
j
Dmax
˚
S
.X/
ˇ
ˇ
X2R
j
D0:
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486 Chapter7
IntegralsofFunctionsofSeveralVariables
Let
U
1
D
˚
j
ˇ
ˇ
R
j
S
and U
2
D
˚
j
ˇ
ˇ
R
j
\S¤;andR
j
\S
c
¤;
:
(7.3.2)
Thentheupperandlowersumsof 
S
overPare
S.P/D
X
j2U
1
V.R
j
/C
X
j2U
2
V.R
j
/
DtotalcontentofthesubrectanglesinPthatintersectS
(7.3.3)
and
s.P/D
X
j2U
1
V.R
j
/
DtotalcontentofthesubrectanglesinP containedinS:
(7.3.4)
Therefore,
S.P/s.P/D
X
j2U
2
V.R
j
/;
whichisthetotalcontentofthesubrectanglesinP thatintersectbothS andS
c
. Since
thesesubrectanglescontain@S,whichdoesnothavezerocontent,thereisan
0
>0such
that
S.P/s.P/
0
foreverypartitionPofR.ByTheorem7.1.12,thisimpliesthat 
S
isnotintegrableonR,
soSisnotJordanmeasurable.
Theorems7.1.19and7.3.1implythefollowingcorollary.
Corollary7.3.2
Iff isboundedandcontinuousonaboundedJordanmeasurableset
S;thenf isintegrableonS:
Lemma7.3.3
SupposethatKisaboundedsetwithzerocontentand;>0:Then
therearecubesC
1
;C
2
;...;C
r
withedgelengths<suchthatC
j
\K¤;;1j r;
K
[r
jD1
C
j
;
(7.3.5)
and
Xr
jD1
V.C
j
/<:
Proof
SinceV.K/D0,
Z
C
K
.X/dXD0
ifCisanycubecontainingK.Fromthisandthedefinitionoftheintegral,thereisaı>0
suchthatifP isanypartitionofCwithkPkıandisanyRiemannsumof 
K
over
P,then
0:
(7.3.6)
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Section7.3
ChangeofVariablesinMultipleIntegrals
487
NowsupposethatP DfC
1
;C
2
;:::;C
k
gisapartitionofCintocubeswith
kPk<min.;ı/;
(7.3.7)
andletC
1
,C
2
,...,C
k
benumberedsothatC
j
\K¤;if1j randC
j
\KD;
ifrC1j k.Then(7.3.5)holds,andatypicalRiemannsumof 
K
overP isofthe
form
D
Xr
jD1
K
.X
j
/V.C
j
/
withX
j
2C
j
,1j r.Inparticular,wecanchooseX
j
fromK,sothat 
K
.X
j
/D1,
and
D
r
X
jD1
V.C
j
/:
Now(7.3.6)and(7.3.7)implythatC
1
,C
2
,...,C
r
havetherequiredproperties.
TransformationsofJordan-MeasurableSets
Toformulatethetheoremonchangeofvariablesinmultipleintegrals,wemustfirstcon-
siderthequestionofpreservationofJordanmeasurabilityunderaregulartransformation.
Lemma7.3.4
SupposethatGWR
n
!R
n
iscontinuouslydifferentiableonabounded
opensetS;andletK beaclosedsubsetofS withzerocontent:ThenG.K/haszero
content.
Proof
SinceKisacompactsubsetoftheopensetS,thereisa
1
>0suchthatthe
compactset
K
1
D
˚
X
ˇ
ˇ
dist.X;K/
1
iscontainedinS(Exercise5.1.26).FromLemma6.2.7,thereisaconstantMsuchthat
jG.Y/G.X/jMjYXj if X;Y2K
1
:
(7.3.8)
Nowsupposethat > 0. SinceV.K/ D D 0,therearecubesC
1
,C
2
,...,C
r
withedge
lengthss
1
,s
2
,...,s
r
<
1
=
p
nsuchthatC
j
\K¤;,1j r,
K
[r
jD1
C
j
;
and
Xr
jD1
V.C
j
/<
(7.3.9)
(Lemma7.3.3).For1j r,letX
j
2C
j
\K.IfX2C
j
,then
jXX
j
js
j
p
n<
1
;
488 Chapter7
IntegralsofFunctionsofSeveralVariables
soX2KandjG.X/G.X
j
/jMjXX
j
jM
p
ns
j
,from(7.3.8).Therefore,G.C
j
/
iscontainedinacube
e
C
j
withedgelength2M
p
ns
j
,centeredatG.X
j
/.Since
V.
e
C
j
/D.2M
p
n/
n
s
n
j
D.2M
p
n/
n
V.C
j
/;
wenowseethat
G.K/
[r
jD1
e
C
j
and
Xr
jD1
V.
e
C
j
/.2M
p
n/
n
Xr
jD1
V.C
j
/<.2M
p
n/
n
;
wherethelastinequalityfollowsfrom(7.3.9). Since.2M
p
n/doesnotdependon,it
followsthatV.G.K//D0.
Theorem7.3.5
SupposethatGWR
n
!R
n
isregularonacompactJordanmeasur-
ablesetS:ThenG.S/iscompactandJordanmeasurable:
Proof
WeleaveittoyoutoprovethatG.S/iscompact(Exercise6.2.23). SinceS
isJordanmeasurable, V.@S/ D 0, , byTheorem7.3.1. Therefore, , V.G.@S// D 0, , by
Lemma 7.3.4. ButG.@S/ D @.G.S//(Exercise6.3.23), , soV.@.G.S/// D D 0, , which
impliesthatG.S/isJordanmeasurable,againbyTheorem7.3.1.
Change ofContentUnderaLinearTransformation
Tomotivateandprovetheruleforchangeofvariablesinmultipleintegrals,wemustknow
howV.L.S//isrelatedtoV.S/ifSisacompactJordanmeasurablesetandLisanonsin-
gularlineartransformation.(FromTheorem7.3.5,L.S/iscompactandJordanmeasurable
inthiscase.) Thenextlemmafromlinearalgebrawillhelptoestablishthisrelationship.
Weomittheproof.
Lemma7.3.6
AnonsingularnnmatrixAcanbewrittenas
ADE
k
E
k1
E
1
;
(7.3.10)
whereeachE
i
isamatrixthatcanbeobtainedfromthennidentitymatrixIbyoneof
thefollowingoperationsW
(a)
interchangingtworowsofII
(b)
multiplyingarowofIbyanonzeroconstantI
(c)
addingamultipleofonerowofItoanother:
Matricesofthekinddescribedinthislemmaarecalledelementarymatrices.Thekeyto
theproofofthelemmaisthatifEisanelementarynnmatrixandAisanynnmatrix,
thenEAisthematrixobtainedbyapplyingtoAthesameoperationthatmustbeapplied
toItoproduceE(Exercise7.3.6). Also,theinverseofanelementarymatrixoftype
(a)
,
(b)
,or
(c)
isanelementarymatrixofthesametype(Exercise7.3.7).
Thenextexampleillustratestheprocedureforfindingthefactorization(7.3.10).
Section7.3
ChangeofVariablesinMultipleIntegrals
489
Example7.3.1
Thematrix
AD
2
4
0 1 1
1 0 1
2 2 0
3
5
isnonsingular,sincedet.A/D4.InterchangingthefirsttworowsofAyields
A
1
D
2
4
1 0 1
0 1 1
2 2 0
3
5
D
b
E
1
A;
where
b
E
1
D
2
4
0 1 0
1 0 0
0 0 1
3
5
:
SubtractingtwicethefirstrowofA
1
fromthethirdyields
A
2
D
2
4
1 0
1
0 1
1
0 2 2
3
5
D
b
E
2
b
E
1
A;
where
b
E
2
D
2
4
1 0 0
0 1 0
2 0 1
3
5
:
SubtractingtwicethesecondrowofA
2
fromthethirdyields
A
3
D
2
4
1 0
1
0 1
1
0 0 4
3
5
D
b
E
3
b
E
2
b
E
1
A;
where
b
E
3
D
2
4
1
0 0
0
1 0
0 2 1
3
5
:
MultiplyingthethirdrowofA
3
by
1
4
yields
A
4
D
2
4
1 0 1
0 1 1
0 0 1
3
5
D
b
E
4
b
A
3
b
E
2
b
E
1
A;
where
b
E
4
D
2
4
1 0
0
0 1
0
0 0 
1
4
3
5
:
490 Chapter7
IntegralsofFunctionsofSeveralVariables
SubtractingthethirdrowofA
4
fromthefirstyields
A
5
D
2
4
1 0 0
0 1 1
0 0 1
3
5
D
b
E
5
b
A
4
b
E
3
b
E
2
b
E
1
A;
where
b
E
5
D
2
4
1 0 1
0 1
0
0 0
1
3
5
:
Finally,subtractingthethirdrowofA
5
fromthesecondyields
I D
b
E
6
b
E
5
b
E
4
b
E
3
b
E
2
b
E
1
A;
(7.3.11)
where
b
E
6
D
2
4
1 0
0
0 1 1
0 0
1
3
5
:
From(7.3.11)andTheorem6.1.16,
AD.
b
E
6
b
E
5
b
E
4
b
E
3
b
E
2
b
E
1
/
1
D
b
E
1
1
b
E
1
2
b
E
1
3
b
E
1
4
b
E
1
5
b
E
1
6
:
Therefore,
ADE
6
E
5
E
4
E
3
E
2
E
1
;
where
E
1
D
b
E
1
6
D
2
4
1 0 0
0 1 1
0 0 1
3
5
,
E
2
D
b
E
1
5
D
2
4
1 0 1
0 1 0
0 0 1
3
5
,
E
3
D
b
A
1
4
D
2
4
1 0
0
0 1
0
0 0 4
3
5
, E
4
D
b
E
1
3
D
2
4
1 0 0
0 1 0
0 2 1
3
5
,
E
5
D
b
E1
2
D
2
4
1 0 0
0 1 0
2 0 1
3
5
,
E
6
D
b
E1
1
D
2
4
0 1 0
1 0 0
0 0 1
3
5
(Exercise7.3.7
(c)
).
Lemma7.3.6andTheorem6.1.7
(c)
implythatanarbitraryinvertiblelineartransforma-
tionLWR
n
!R
n
,definedby
XDL.Y/DAY;
(7.3.12)
canbewrittenasacomposition
LDL
k
ıL
k1
ııL
1
;
(7.3.13)
where
L
i
.Y/DE
i
Y; 1ik:
Section7.3
ChangeofVariablesinMultipleIntegrals
491
Theorem7.3.7
IfSisacompactJordanmeasurablesubsetofR
n
andLWR
n
!R
n
istheinvertiblelineartransformationXDL.Y/DAY;then
V.L.S//Djdet.A/jV.S/:
(7.3.14)
Proof
Theorem7.3.5impliesthatL.S/isJordanmeasurable.If
V.L.R//Djdet.A/jV.R/
(7.3.15)
wheneverR isa rectangle, then(7.3.14)holdsifS isanycompactJordanmeasurable
set. Toseethis, , supposethat > > 0, letRbearectanglecontainingS, andletP D
fR
1
;R
2
;:::;R
k
gbeapartitionofRsuchthattheupperandlowersumsof 
S
overP
satisfytheinequality
S.P/s.P/<:
(7.3.16)
LetU
1
andU
2
beasin(7.3.2).From(7.3.3)and(7.3.4),
s.P/D
X
j2U
1
V.R
j
/V.S/
X
j2U
1
V.R
j
/C
X
j2U
2
V.R
j
/DS.P/:
(7.3.17)
Theorem7.3.7impliesthatL.R
1
/,L.R
2
/,...,L.R
k
/andL.S/areallJordanmeasurable.
Since
[
j2U
1
R
j
S
[
j2S
1
[S
2
R
j
;
itfollowsthat
L
0
@
[
j2U
1
R
j
1
A
L.S/L
0
@
[
j2S
1
[S
2
R
j
1
A
:
SinceLisone-to-oneonR
n
,thisimpliesthat
X
j2U
1
V.L.R
j
//V.L.S//
X
j2U
1
V.L.R
j
//C
X
j2U
2
V.L.R
j
//:
(7.3.18)
Ifweassumethat(7.3.15)holdswheneverRisarectangle,then
V.L.R
j
//Djdet.A/jV.R
j
/; 1j j k;
so(7.3.18)impliesthat
s.P/
V.L.S//
jdet.A/j
S.P/:
This,(7.3.16)and(7.3.17)implythat
ˇ
ˇ
ˇ
ˇ
V.S/
V.L.S//
jdet.A/j
ˇ
ˇ
ˇ
ˇ
<I
hence,sincecanbemadearbitrarilysmall,(7.3.14)followsforanyJordanmeasurable
set.
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