pdfsharp c# : Add links to pdf online Library software component .net winforms wpf mvc TRENCH_REAL_ANALYSIS47-part267

462 Chapter7
IntegralsofFunctionsofSeveralVariables
(b)
Use
(a)
toshowthatiff iscontinuousonRandP P isapartitionofR,then
thereisaRiemannsumoff overP P thatequals
R
R
f.X/dX.
29.
Supposethatf iscontinuouslydifferentiableonarectangleR.Showthatthereisa
constantMsuchthat
ˇ
ˇ
ˇ
ˇ

Z
R
f.X/dX
ˇ
ˇ
ˇ
ˇ
MkPk
ifisanyRiemannsumoff overapartitionPofR.H
INT
:UseExercise7.1.28.b/
andTheorem5.4.5:
30.
Supposethat
R
b
a
f.x/dxand
R
d
c
g.y/dyexist,andletRDŒa;bŒc;d.
(a)
UseTheorems3.2.7and7.1.12toshowthat
Z
R
f.x/d.x;y/ and
Z
R
g.y/d.x;y/
bothexist.
(b)
UseTheorem7.1.27toprovethat
R
R
f.x/g.y/d.x;y/exists.
(c)
JustifyusingtheargumentgiveninExercise7.1.3toshowthat
Z
R
f.x/g.y/d.x;y/D
Z
b
a
f.x/dx
Z
d
c
g.y/dy
!
:
31. (a)
Supposethatf isintegrableonSandS
0
isobtainedbyremovingasetof
zerocontentfromS. Showthatf isintegrableonS
0
and
R
S
0
f.X/dX D
R
S
f.X/dX.
(b)
ProveCorollary7.1.31.
7.2ITERATEDINTEGRALSANDMULTIPLEINTEGRALS
Exceptforverysimpleexamples, itisimpracticaltoevaluatemultipleintegralsdirectly
fromDefinitions7.1.2and7.1.17.Fortunately,thiscanusuallybeaccomplishedbyevalu-
atingnsuccessiveordinaryintegrals.Tomotivatethemethod,letusfirstassumethatf is
continuousonRDŒa;bŒc;d. Then,foreachyinŒc;d,f.x;y/iscontinuouswith
respecttoxonŒa;b,sotheintegral
F.y/D
Z
b
a
f.x;y/dx
exists. Moreover,theuniformcontinuityoff onRimpliesthatF F iscontinuous(Exer-
cise7.2.3)andthereforeintegrableonŒc;d.Wesaythat
I
1
D
Z
d
c
F.y/dyD
Z
d
c
Z
b
a
f.x;y/dx
!
dy
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Section7.2
IteratedIntegralsandMultipleIntegrals
463
isaniteratedintegraloffoverR.Wewillusuallywriteitas
I
1
D
Z
d
c
dy
Z
b
a
f.x;y/dx:
Anotheriteratedintegralcanbedefinedbywriting
G.x/D
Z
d
c
f.x;y/dy; axb;
anddefining
I
2
D
Z
b
a
G.x/dxD
Z
b
a
Z
d
c
f.x;y/dy
!
dx;
whichweusuallywriteas
I
2
D
Z
b
a
dx
Z
d
c
f.x;y/dy:
Example7.2.1
Let
f.x;y/DxCy
andRDŒ0;1Œ1;2.Then
F.y/D
Z
1
0
f.x;y/dxD
Z
1
0
.xCy/dxD
x
2
2
Cxy
ˇ
ˇ
ˇ
ˇ
1
xD0
D
1
2
Cy
and
I
1
D
Z
2
1
F.y/dyD
Z
2
1
1
2
Cy
dyD
y
2
C
y2
2
ˇ
ˇ
ˇ
ˇ
2
1
D2:
Also,
G.x/D
Z
2
1
.xCy/dyD
xyC
y
2
2
ˇ
ˇ
ˇ
ˇ
2
yD1
D.2xC2/
xC
1
2
DxC
3
2
;
and
I
2
D
Z
1
0
G.x/dxD
Z
1
0
xC
3
2
dxD
x
2
2
C
3x
2
ˇ
ˇ
ˇ
ˇ
1
0
D2:
Inthisexample,I
1
D I
2
;moreover,onsettingaD 0,b D1,c D1,andd d D D 2in
Example7.1.1,weseethat
Z
R
.xCy/d.x;y/D2;
sothecommonvalueoftheiteratedintegralsequalsthemultipleintegral. Thefollowing
theoremshowsthatthisisnotanaccident.
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464 Chapter7
IntegralsofFunctionsofSeveralVariables
Theorem7.2.1
Supposethatf isintegrableonRDŒa;bŒc;dand
F.y/D
Z
b
a
f.x;y/dx
existsforeachyinŒc;d:ThenFisintegrableonŒc;d;and
Z
d
c
F.y/dyD
Z
R
f.x;y/d.x;y/I
(7.2.1)
thatis;
Z
d
c
dy
Z
b
a
f.x;y/dxD
Z
R
f.x;y/d.x;y/:
(7.2.2)
Proof
Let
P
1
WaDx
0
<x
1
<<x
r
Db and P
2
WcDy
0
<y
1
<<y
s
Dd
bepartitionsofŒa;bandŒc;d,andPDP
1
P
2
.Supposethat
y
j1

j
y
j
; 1j j s;
(7.2.3)
so
D
Xs
jD1
F.
j
/.y
j
y
j1
/
(7.2.4)
isatypicalRiemannsumofFoverP
2
.Since
F.
j
/D
Z
b
a
f.x;
j
/dxD
Xr
iD1
Z
x
x
i1
f.x;
j
/dx;
(7.2.3)impliesthatif
m
ij
Dinf
˚
f.x;y/
ˇ
ˇ
x
i1
xx
i
;y
j1
yy
j
and
M
ij
Dsup
˚
f.x;y/
ˇ
ˇ
x
i1
xx
i
;y
j1
yy
j
;
then
Xr
iD1
m
ij
.x
i
x
i1
/F.
j
/
Xr
iD1
M
ij
.x
i
x
i1
/:
Multiplyingthisbyy
j
y
j1
andsummingfromj D1toj Dsyields
Xs
jD1
Xr
iD1
m
ij
.x
i
x
i1
/.y
j
y
j1
/
Xs
jD1
F.
j
/.y
j
y
j1
/
s
X
jD1
r
X
iD1
M
ij
.x
i
x
i1
/.y
j
y
j1
/;
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Section7.2
IteratedIntegralsandMultipleIntegrals
465
which,from(7.2.4),canberewrittenas
s
f
.P/S
f
.P/;
(7.2.5)
wheres
f
.P/andS
f
.P/aretheloweranduppersumsoff overP. Nowlets
F
.P
2
/and
S
F
.P
2
/betheloweranduppersumsofFoverP
2
;sincetheyarerespectivelytheinfimum
andsupremumoftheRiemannsumsofF overP
2
(Theorem3.1.4),(7.2.5)impliesthat
s
f
.P/s
F
.P
2
/S
F
.P
2
/S
f
.P/:
(7.2.6)
Sincef isintegrableonR,thereisforeach>0apartitionPofRsuchthatS
f
.P/
s
f
.P/ <,fromTheorem7.1.12. Consequently,from(7.2.6),thereisapartitionP
2
of
Œc;dsuchthatS
F
.P
2
/s
F
.P
2
/<,soFisintegrableonŒc;d,fromTheorem3.2.7.
Itremainstoverify(7.2.1). From(7.2.4)andthedefinitionof
R
d
c
F.y/dy,thereisfor
each>0aı>0suchthat
ˇ
ˇ
ˇ
ˇ
ˇ
Z
d
c
F.y/dy
ˇ
ˇ
ˇ
ˇ
ˇ
< if kP
2
k<ıI
thatis,
<
Z
d
c
F.y/dy<C if kP
2
k<ı:
Thisand(7.2.5)implythat
s
f
.P/<
Z
d
c
F.y/dy<S
f
.P/C if kPk<ı;
andthisimpliesthat
Z
R
f.x;y/d.x;y/
Z
d
c
F.y/dy
Z
R
f.x;y/d.x;y/C
(7.2.7)
(Definition7.1.4).Since
Z
R
f.x;y/d.x;y/D
Z
R
f.x;y/d.x;y/
(Theorem7.1.8)andcanbemadearbitrarilysmall,(7.2.7)implies(7.2.1).
IffiscontinuousonR,thenfsatisfiesthehypothesesofTheorem7.2.1(Exercise7.2.3),
so(7.2.2)isvalidinthiscase.
If
R
R
f.x;y/d.x;y/and
Z
d
c
f.x;y/dy; axb;
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466 Chapter7
IntegralsofFunctionsofSeveralVariables
exist,thenbyinterchangingxandyinTheorem7.2.1,weseethat
Z
b
a
dx
Z
d
c
f.x;y/dyD
Z
R
f.x;y/d.x;y/:
Thisand(7.2.2)yieldthefollowingcorollaryofTheorem7.2.1.
Corollary7.2.2
Iff isintegrableonŒa;bŒc;d;then
Z
b
a
dx
Z
d
c
f.x;y/dyD
Z
d
c
dy
Z
b
a
f.x;y/dx;
providedthat
R
d
c
f.x;y/dyexistsforaxband
R
b
a
f.x;y/dxexistsforcyd:
Inparticular;thesehypothesesholdiff iscontinuousonŒa;bŒc;d:
Example7.2.2
Thefunction
f.x;y/DxCy
iscontinuouseverywhere,so(7.2.2)holdsforeveryrectangleR. Forexample,letR R D
Œ0;1Œ1;2.Then(7.2.2)yields
Z
R
.xCy/d.x;y/D
Z
2
1
dy
Z
1
0
.xCy/dxD
Z
2
1
"
x2
2
Cxy
ˇ
ˇ
ˇ
ˇ
1
xD0
#
dy
D
Z
2
1
1
2
Cy
dyD
y
2
C
y2
2
ˇ
ˇ
ˇ
ˇ
2
1
D2:
Sincef alsosatisfiesthehypothesesofTheorem7.2.1withx x andyinterchanged, we
cancalculatethedoubleintegralfromtheiteratedintegralinwhichtheintegrationsare
performedintheoppositeorder;thus,
Z
R
.xCy/d.x;y/D
Z
1
0
dx
Z
2
1
.xCy/dyD
Z
1
0
"
xyC
y
2
2
ˇ
ˇ
ˇ
ˇ
2
yD1
#
dx
D
Z
1
0
xC
3
2
dxD
x
2
2
C
3x
2
ˇ
ˇ
ˇ
ˇ
1
0
D2:
AplausiblepartialconverseofTheorem7.2.1wouldbethatif
R
d
c
dy
R
b
a
f.x;y/dx
existsthensodoes
R
R
f.x;y/d.x;y/;however,thenextexampleshowsthatthisneednot
beso.
Example7.2.3
Iff isdefinedonRDŒ0;1Œ0;1by
f.x;y/D
2xy ifyisrational;
y
ifyisirrational;
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Section7.2
IteratedIntegralsandMultipleIntegrals
467
then
Z
1
0
f.x;y/dxDy; 0y1;
and
Z
1
0
dy
Z
1
0
f.x;y/dxD
Z
1
0
ydyD
1
2
:
However,f isnotintegrableonR(Exercise7.2.7).
ThenexttheoremgeneralizesTheorem7.2.1toR
n
.
Theorem7.2.3
LetI
1
;I
2
;...;I
n
beclosedintervalsandsupposethatfisintegrable
onRDI
1
I
2
I
n
:Supposethatthereisanintegerpinf1;2;:::;n1gsuchthat
F
p
.x
pC1
;x
pC2
;:::;x
n
/D
Z
I
1
I
2
I
p
f.x
1
;x
2
;:::;x
n
/d.x
1
;x
2
;:::;x
p
/
existsforeach.x
pC1
;x
pC2
;:::;x
n
/inI
pC1
I
pC2
I
n
:Then
Z
I
pC1
I
pC2
I
n
F
p
.x
pC1
;x
pC2
;:::;x
n
/d.x
pC1
;x
pC2
;:::;x
n
/
existsandequals
R
R
f.X/dX.
Proof
Forconvenience,denote.x
pC1
;x
pC2
;:::;x
n
/byY.Denote
b
RDI
1
I
2

I
p
andT DI
pC1
I
pC2
I
n
.Let
b
PDf
b
R
1
;
b
R
2
;:::;
b
R
k
gandQDfT
1
;T
2
;:::;T
s
g
bepartitionsof
b
RandT,respectively.Thenthecollectionofrectanglesoftheform
b
R
i
T
j
(1i k,1j j s)isapartitionPofR;moreover,everypartitionPofRisofthis
form.
Supposethat
Y
j
2T
j
; 1j j s;
(7.2.8)
so
D
Xs
jD1
F
p
.Y
j
/V.T
j
/
(7.2.9)
isatypicalRiemannsumofF
p
overQ.Since
F
p
.Y
j
/D
Z
b
R
f.x
1
;x
2
;:::;x
p
;Y
j
/d.x
1
;x
2
;:::;x
p
/
D
Xk
jD1
Z
b
R
j
f.x
1
;x
2
;:::;x
p
;Y
j
/d.x
1
;x
2
;:::;x
p
/;
(7.2.8)impliesthatif
m
ij
Dinf
n
f.x
1
;x
2
;:::;x
p
;Y/
ˇ
ˇ
.x
1
;x
2
;:::;x
p
/2
b
R
i
;Y2T
j
o
and
M
ij
Dsup
n
f.x
1
;x
2
;:::;x
p
;Y/
ˇ
ˇ
.x
1
;x
2
;:::;x
p
/2
b
R
i
;Y2T
j
o
;
468 Chapter7
IntegralsofFunctionsofSeveralVariables
then
k
X
iD1
m
ij
V.
b
R
i
/F
p
.Y
j
/
k
X
iD1
M
ij
V.
b
R
i
/:
MultiplyingthisbyV.T
j
/andsummingfromj D1toj Dsyields
Xs
jD1
Xk
iD1
m
ij
V.
b
R
i
/V.T
j
/
Xs
jD1
F
p
.Y
j
/V.T
j
/
Xs
jD1
Xk
iD1
M
ij
V.
b
R
i
/V.T
j
/;
which,from(7.2.9),canberewrittenas
s
f
.P/S
f
.P/;
(7.2.10)
wheres
f
.P/andS
f
.P/aretheloweranduppersumsoff overP. . Nowlets
F
p
.Q/and
S
F
p
.Q/betheloweranduppersumsofF
p
overQ;sincetheyarerespectivelytheinfimum
andsupremumoftheRiemannsumsofF
p
overQ(Theorem7.1.5),(7.2.10)impliesthat
s
f
.P/s
F
p
.Q/S
F
p
.Q/S
f
.P/:
(7.2.11)
Sincef isintegrableonR,thereisforeach>0apartitionPofRsuchthatS
f
.P/
s
f
.P/<,fromTheorem7.1.12.Consequently,from(7.2.11),thereisapartitionQofT
suchthatS
F
p
.Q/s
F
p
.Q/<,soF
p
isintegrableonT,fromTheorem7.1.12.
Itremainstoverifythat
Z
R
f.X/dXD
Z
T
F
p
.Y/dY:
(7.2.12)
From(7.2.9)andthedefinitionof
R
T
F
p
.Y/dY,thereisforeach>0aı>0suchthat
ˇ
ˇ
ˇ
ˇ
Z
T
F
p
.Y/dY
ˇ
ˇ
ˇ
ˇ
< if kQk<ıI
thatis,
<
Z
T
F
p
.Y/dY<C if kQk<ı:
Thisand(7.2.10)implythat
s
f
.P/<
Z
T
F
p
.Y/dY<S
f
.P/C if kPk<ı;
andthisimpliesthat
Z
R
f.X/dX
Z
T
F
p
.Y/dY
Z
R
f.X/dXC:
(7.2.13)
Since
Z
R
f.X/dXD
Z
R
f.X/dX(Theorem7.1.8)andcanbemadearbitrarilysmall,
(7.2.13)implies(7.2.12).
Section7.2
IteratedIntegralsandMultipleIntegrals
469
Theorem7.2.4
LetI
j
D Œa
j
;b
j
;1j n,andsupposethatf isintegrableon
RDI
1
I
2
I
n
:Supposealsothattheintegrals
F
p
.x
pC1
;:::;x
n
/D
Z
I
1
I
2
I
p
f.X/d.x
1
;x
2
;:::;x
p
/; 1pn1;
existforall
.x
pC1
;:::;x
n
/ in I
pC1
I
n
:
Thentheiteratedintegral
Z
b
n
a
n
dx
n
Z
b
n1
a
n1
dx
n1

Z
b
2
a
2
dx
2
Z
b
1
a
1
f.X/dx
1
existsandequals
R
R
f.X/dX:
Proof
Theproofisbyinduction.FromTheorem7.2.1,thepropositionistruefornD2.
Nowassumen>2andthepropositionistruewithnreplacedbyn1.Holdingx
n
fixed
andapplyingthisassumptionyields
F
n
.x
n
/D
Z
b
n1
a
n1
dx
n1
Z
b
n2
a
n2
dx
n2

Z
b
2
a
2
dx
2
Z
b
1
a
1
f.X/dx
1
:
NowTheorem7.2.3withpDn1completestheinduction.
Example7.2.4
LetRDŒ0;1Œ1;2Œ0;1and
f.x;y;´/DxCyC´:
Then
F
1
.y;´/D
Z
1
0
.xCyC´/dxD
x
2
2
CxyCx´
ˇ
ˇ
ˇ
ˇ
1
xD0
D
1
2
CyC´;
F
2
.´/D
Z
2
1
F
1
.y;´/dyD
Z
2
1
1
2
CyC´
dy
D
y
2
C
y
2
2
Cy´
ˇ
ˇ
ˇ
ˇ
2
yD1
D2C´;
and
Z
R
f.x;y;´/d.x;y;´/D
Z
1
0
F
2
.´/d´D
Z
1
0
.2C´/d´D
2´C
´
2
2
ˇ
ˇ
ˇ
ˇ
1
0
D
5
2
:
ThehypothesesofTheorems7.2.3and 7.2.4arestatedsoastojustifysuccessivein-
tegrationswithrespecttox
1
,thenx
2
,thenx
3
,andsoforth. Itislegitimatetouseother
ordersofintegrationifthehypothesesareadjustedaccordingly.Forexample,supposethat
470 Chapter7
IntegralsofFunctionsofSeveralVariables
fi
1
;i
2
;:::;i
n
gisapermutationoff1;2;:::;ngand
R
R
f.X/dXexists,alongwith
Z
I
i
1
I
i
2
I
i
j
f.X/d.x
i
1
;x
i
2
;:::;x
i
j
/; 1j j n1;
(7.2.14)
foreach
.x
i
jC1
;x
i
jC2
;:::;x
i
n
/ in I
i
jC1
I
i
jC2
I
i
n
:
(7.2.15)
Then,byrenamingthevariables,weinferfromTheorem7.2.4that
Z
R
f.X/dXD
Z
b
i
n
a
i
n
dx
i
n
Z
b
i
n1
a
i
n1
dx
i
n1

Z
b
i
2
a
i
2
dx
i
2
Z
b
i
1
a
i
1
f.X/dx
i
1
:
(7.2.16)
SincetherearenŠpermutationsoff1;2;:::;ng,therearenŠwaysofevaluatingamul-
tipleintegraloverarectangleinR
n
,providedthattheintegrandsatisfiesappropriatehy-
potheses. Inparticular,iff iscontinuousonRandfi
1
;i
2
;:::;i
n
gisanypermutationof
f1;2;:::;ng,thenfiscontinuouswithrespectto.x
i
1
;x
i
2
;:::;x
i
j
/onI
i
1
I
i
2
I
i
j
foreachfixed.x
i
jC1
;x
i
jC2
;:::;x
i
n
/satisfying(7.2.15).Therefore,theintegrals(7.2.14)
existforeverypermutationoff1;2;:::;ng(Theorem7.1.13). Wesummarizethisinthe
nexttheorem,whichnowfollowsfromTheorem7.2.4.
Theorem7.2.5
Iff iscontinuouson
RDŒa
1
;b
1
Œa
2
;b
2
Œa
n
;b
n
;
then
R
R
f.X/dXcanbeevaluatedbyiteratedintegralsinanyofthenŠwaysindicatedin
(7.2.16):
Example7.2.5
Iff iscontinuousonRDŒa
1
;b
1
Œa
2
;b
2
Œa
3
;b
3
,then
Z
R
f.x;y;´/d.x;y;´/D
Z
b
3
a
3
Z
b
2
a
2
dy
Z
b
1
a
1
f.x;y;´/dx
D
Z
b
2
a
2
dy
Z
b
3
a
3
Z
b
1
a
1
f.x;y;´/dx
D
Z
b
3
a
3
Z
b
1
a
1
dx
Z
b
2
a
2
f.x;y;´/dy
D
Z
b
1
a
1
dx
Z
b
3
a
3
Z
b
2
a
2
f.x;y;´/dy
D
Z
b
2
a
2
dy
Z
b
1
a
1
dx
Z
b
3
a
3
f.x;y;´/d´
D
Z
b
1
a
1
dx
Z
b
2
a
2
dy
Z
b
3
a
3
f.x;y;´/d´:
Section7.2
IteratedIntegralsandMultipleIntegrals
471
IntegralsoverMoreGeneralSets
Wenowconsidertheproblemofevaluatingmultipleintegralsovermoregeneralsets.First,
supposethatf isintegrableonasetoftheform
SD
˚
.x;y/
ˇ
ˇ
u.y/xv.y/; cyd
(7.2.17)
(Figure7.2.1).
Ifu.y/aandv.y/bforcyd,and
f
S
.x;y/D
(
f.x;y/; .x;y/2S;
0;
.x;y/62S;
(7.2.18)
then
Z
S
f.x;y/d.x;y/D
Z
R
f
S
.x;y/d.x;y/;
whereRDŒa;bŒc;d..FromTheorem7.2.1,
Z
R
f
S
.x;y/d.x;y/D
Z
d
c
dy
Z
b
a
f
S
.x;y/dx
providedthat
R
b
a
f
S
.x;y/dx existsforeachyinŒc;d. From(7.2.17)and(7.2.18),this
integralcanbewrittenas
Z
v.y/
u.y/
f.x;y/dx:
(7.2.19)
Thus,wehaveprovedthefollowingtheorem.
y
x
b
a
x = v(y)
y = c
y = d
x = u(y)
Figure7.2.1
Theorem7.2.6
Iff isintegrableonthesetS S in(7.2.17)andtheintegral(7.2.19)
existsforcyd;then
Z
S
f.x;y/d.x;y/D
Z
d
c
dy
Z
v.y/
u.y/
f.x;y/dx:
(7.2.20)
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