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442 Chapter7
IntegralsofFunctionsofSeveralVariables
NowsupposethatXsatisfies(7.1.12),andconsidertheRiemannsum
0
D
Xn
jD1
f.X
0
j
/V.R
j
/
overthesamepartitionP,where
X
0
j
D
X
j
; j j ¤i;
X; j j Di:
Since
j
0
jDjf.X/f.X
i
/jV.R
i
/;
(7.1.12)implies(7.1.11).
BecauseofTheorem7.1.3,weneedconsideronlyboundedfunctionsinconnectionwith
Definition7.1.2. Asinthecasewheren n D D 1,itisnowconvenienttodefinetheupper
andlowerintegralsofaboundedfunctionoverarectangle. Thefollowingdefinitionis
analogoustoDefinition3.1.3.
Definition7.1.4
Iff isboundedonarectangleRinR
n
andP DfR
1
;R
2
;:::;R
k
g
isapartitionofR,let
M
j
D sup
X2R
j
f.X/; m
j
D inf
X2R
j
f.X/:
Theuppersumoff overPis
S.P/D
Xk
jD1
M
j
V.R
j
/;
andtheupperintegraloffoverR,denotedby
Z
R
f.X/dX;
istheinfimumofalluppersums.Thelowersumoff overP P is
s.P/D
Xk
jD1
m
j
V.R
j
/;
andthelowerintegraloff overR,denotedby
Z
R
f.X/dX;
isthesupremumofalllowersums.
ThefollowingtheoremisanalogoustoTheorem3.1.4.
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Section7.1
DefinitionandExistenceoftheMultipleIntegral
443
Theorem7.1.5
Letf beboundedonarectangleRandletPbeapartitionofR:
Then
(a)
TheuppersumS.P/off overPisthesupremumofthesetofallRiemannsumsof
f overP:
(b)
Thelowersums.P/off overPistheinfimumofthesetofallRiemannsumsoff
overP:
Proof
Exercise7.1.5.
If
mf.X/M
forXinR;
then
mV.R/s.P/S.P/MV.R/I
therefore,
R
R
f.X/dXand
R
R
f.X/dXexist,areunique,andsatisfytheinequalities
mV.R/
Z
R
f.X/dXMV.R/
and
mV.R/
Z
R
f.X/dXMV.R/:
Theupperandlowerintegralsarealsowrittenas
Z
R
f.x;y/d.x;y/ and
Z
R
f.x;y/d.x;y/ .nD2/;
Z
R
f.x;y;´/d.x;y;´/ and
Z
R
f.x;y;´/d.x;y;´/ .nD3/;
or
Z
R
f.x
1
;x
2
;:::;x
n
/d.x
1
;x
2
;:::;x
n
/
and
Z
R
f.x
1
;x
2
;:::;x
n
/d.x
1
;x
2
;:::;x
n
/
(narbitrary):
Example7.1.2
Find
R
R
f.x;y/d.x;y/and
R
R
f.x;y/d.x;y/,withR D D Œa;b
Œc;dand
f.x;y/DxCy;
asinExample7.1.1.
Solution
LetP
1
andP
2
bepartitionsofŒa;bandŒc;d;thus,
P
1
WaDx
0
<x
1
<<x
r
Db and
P
2
WcDy
0
<y
1
<<y
s
Dd:
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444 Chapter7
IntegralsofFunctionsofSeveralVariables
ThemaximumandminimumvaluesoffontherectangleŒx
i1
;x
i
Œy
j1
;y
j
arex
i
Cy
j
andx
i1
Cy
j1
,respectively.Therefore,
S.P/D
Xr
iD1
Xs
jD1
.x
i
Cy
j
/.x
i
x
i1
/.y
j
y
j1
/
(7.1.13)
and
s.P/D
Xr
iD1
Xs
jD1
.x
i1
Cy
j1
/.x
i
x
i1
/.y
j
y
j1
/:
(7.1.14)
Bysubstituting
x
i
Cy
j
D
1
2
Œ.x
i
Cx
i1
/C.y
j
Cy
j1
/C.x
i
x
i1
/C.y
j
y
j1
/
into(7.1.13),wefindthat
S.P/D
1
2
.†
1
C†
2
C†
3
C†
4
/;
(7.1.15)
where
1
D
Xr
iD1
.x
2
i
x
2
i1
/
Xs
jD1
.y
j
y
j1
/ D.b
2
a
2
/.dc/;
2
D
r
X
iD1
.x
i
x
i1
/
s
X
jD1
.y
2
j
y
2
j1
/ D.ba/.d2c2/;
3
D
Xr
iD1
.x
i
x
i1
/
2
Xs
jD1
.y
j
y
j1
/kPk.ba/.dc/;
4
D
Xr
iD1
.x
i
x
i1
/
Xs
jD1
.y
j
y
j1
/
2
kPk.ba/.dc/:
Substitutingthesefourresultsinto(7.1.15)showsthat
I<S.P/<ICkPk.ba/.d c/;
where
ID
.dc/.b
2
a
2
/C.ba/.d
2
c
2
/
2
:
Fromthis,weseethat
Z
R
.xCy/d.x;y/DI:
Aftersubstituting
x
i1
Cy
j1
D
1
2
Œ.x
i
Cx
i1
/C.y
j
Cy
j1
/.x
i
x
i1
/.y
j
y
j1
/
into(7.1.14),asimilarargumentshowsthat
IkPk.ba/.dc/<s.P/<I;
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Section7.1
DefinitionandExistenceoftheMultipleIntegral
445
so
Z
R
.xCy/d.x;y/DI:
WenowproveananalogofLemma3.2.1.
Lemma7.1.6
Supposethatjf.X/jMifXisintherectangle
RDŒa
1
;b
1
Œa
2
;b
2
Œa
n
;b
n
:
LetP DP
1
P
2
P
n
andP
0
DP
0
1
P
0
2
P
0
n
bepartitionsofR;whereP
0
j
isobtainedbyaddingr
j
partitionpointstoP
j
;1j n:Then
S.P/S.P
0
/S.P/2MV.R/
0
@
Xn
jD1
r
j
b
j
a
j
1
A
kPk
(7.1.16)
and
s.P/s.P
0
/s.P/C2MV.R/
0
@
n
X
jD1
r
j
b
j
a
j
1
A
kPk:
(7.1.17)
Proof
Wewillprove(7.1.16)andleavetheproofof(7.1.17)toyou(Exercise7.1.7).
FirstsupposethatP
0
1
isobtainedbyaddingonepointtoP
1
,andP
0
j
DP
j
for2j n.
IfP
r
isdefinedby
P
r
Wa
r
Da
r0
<a
r1
<<a
rm
r
Db
r
; 1rn;
thenatypicalsubrectangleofPisoftheform
R
j
1
j
2
j
n
DŒa
1;j
1
1
;a
1j
1
Œa
2;j
2
1
;a
2j
2
Œa
n;j
n
1
;a
nj
n
:
LetcbetheadditionalpointintroducedintoP
1
toobtainP
0
1
,andsupposethat
a
1;k1
<c<a
1k
:
Ifj
1
¤k,thenR
j
1
j
2
j
n
iscommontoPandP
0
,sothetermsassociatedwithitinS.P
0
/
andS.P/cancelinthedifferenceS.P/S.P
0
/.Toanalyzethetermsthatdonotcancel,
define
R
.1/
kj
2
j
n
DŒa
1;k1
;cŒa
2;j
2
1
;a
2j
2
Œa
n;j
n
1
;a
nj
n
;
R
.2/
kj
2
j
n
DŒc;a
1k
Œa
2;j
2
1
;a
2j
2
Œa
n;j
n
1
;a
nj
n
;
M
kj
2
j
n
Dsup
˚
f.X/
ˇ
ˇ
X2R
kj
2
j
n
(7.1.18)
and
M
.i/
kj
2
j
n
Dsup
n
f.X/
ˇ
ˇ
X2R
.i/
kj
2
j
n
o
; iD1;2:
(7.1.19)
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446 Chapter7
IntegralsofFunctionsofSeveralVariables
ThenS.P/S.P
0
/isthesumoftermsoftheform
h
M
kj
2
j
n
.a
1k
a
1;k1
/M
.1/
kj
2
j
n
.ca
1;k1
/M
.2/
kj
2
j
n
.a
1k
c/
i
.a
2j
2
a
2;j
2
1
/.a
nj
n
a
n;j
n
1
/:
(7.1.20)
Thetermswithinthebracketscanberewrittenas
.M
kj
2
j
n
M
.1/
kj
2
j
n
/.ca
1;k1
/C.M
kj
2
j
n
M
.2/
kj
2
j
n
/.a
1k
c/;
(7.1.21)
whichisnonnegative,becauseof(7.1.18)and(7.1.19).Therefore,
S.P
0
/S.P/:
(7.1.22)
Moreover,thequantityin(7.1.21)isnotgreaterthan2M.a
1k
a
1;k1
/,so(7.1.20)implies
thatthegeneralsurvivingterminS.P/S.P
0
/isnotgreaterthan
2MkPk.a
2j
2
a
2;j
2
1
/.a
nj
n
a
n;j
n
1
/:
Thesumofthesetermsasj
2
,...,j
n
assumeallpossiblevalues1j
i
m
i
,2in,
is
2MkPk.b
2
a
2
/.b
n
a
n
/D
2MkPkV.R/
b
1
a
1
:
Thisimpliesthat
S.P/S.P
0
/C
2MkPkV.R/
b
1
a
1
:
Thisand(7.1.22)imply(7.1.16)forr
1
D1andr
2
DDr
n
D0.
Similarly,ifr
i
D1forsomeiinf1;:::;ngandr
j
D0ifj ¤i,then
S.P/S.P
0
/C
2MkPkV.R/
b
i
a
i
:
Toobtain(7.1.16)inthegeneralcase,repeatthisargumentr
1
Cr
2
CCr
n
times,asin
theproofofLemma3.2.1.
Lemma7.1.6impliesthefollowingtheoremsandlemma,withproofsanalogoustothe
proofsoftheircounterpartsinSection3.2.
Theorem7.1.7
Iff isboundedonarectangleR;then
Z
R
f.X/dX
Z
R
f.X/dX:
Proof
Exercise7.1.8.
ThenexttheoremisanalogoustoTheorem3.2.3.
Theorem7.1.8
Iff isintegrableonarectangleR;then
Z
R
f.X/dXD
Z
R
f.X/dXD
Z
R
f.X/dX:
Proof
Exercise7.1.9.
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Section7.1
DefinitionandExistenceoftheMultipleIntegral
447
Lemma7.1.9
Iff isboundedonarectangleRand>0;thereisaı>0suchthat
Z
R
f.X/dXS.P/<
Z
R
f.X/dXC
and
Z
R
f.X/dXs.P/>
Z
R
f.X/dX
ifkPk<ı:
Proof
Exercise7.1.10.
ThenexttheoremisanalogoustoTheorem3.2.5.
Theorem7.1.10
Iff isboundedonarectangleRand
Z
R
f.X/dXD
Z
R
f.X/dXDL;
thenf isintegrableonR;and
Z
R
f.X/dXDL:
Proof
Exercise7.1.11.
Theorems7.1.8and7.1.10implythefollowingtheorem,whichisanalogoustoTheo-
rem3.2.6.
Theorem7.1.11
Aboundedfunctionf isintegrableonarectangleRifandonlyif
Z
R
f.X/dXD
Z
R
f.X/dX:
Thenexttheoremtranslatesthisintoatestthatcanbeconvenientlyapplied.Itisanalo-
goustoTheorem3.2.7.
Theorem7.1.12
Iff isboundedonarectangleR;thenf isintegrableonRifand
onlyifforevery>0thereisapartitionP ofRsuchthat
S.P/s.P/<:
Proof
Exercise7.1.12.
Theorem7.1.12providesausefulcriterionforintegrability. Thenexttheoremis s an
importantapplication.ItisanalogoustoTheorem3.2.8.
Theorem7.1.13
IffiscontinuousonarectangleRinR
n
;thenfisintegrableonR:
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448 Chapter7
IntegralsofFunctionsofSeveralVariables
Proof
Let>0. Sincef isuniformlycontinuousonR(Theorem5.2.14),thereisa
ı>0suchthat
jf.X/f.X
0
/j<
V.R/
(7.1.23)
ifXandX
0
areinRandjXX
0
j<ı.LetP DfR
1
;R
2
;:::;R
k
gbeapartitionofRwith
kPk<ı=
p
n.Sincef iscontinuousonR,therearepointsX
j
andX
0
j
inR
j
suchthat
f.X
j
/DM
j
D sup
X2R
j
f.X/ and
f.X
0
j
/Dm
j
D inf
X2R
j
f.X/
(Theorem5.2.12).Therefore,
S.P/s.P/D
Xn
jD1
.f.X
j
/f.X
0
j
//V.R
j
/:
SincekPk<ı=
p
n,jX
j
X0
j
j<ı,and,from(7.1.23)withXDX
j
andXDX0
j
,
S.P/s.P/<
V.R/
Xk
jD1
V.R
j
/D:
Hence,fisintegrableonR,byTheorem7.1.12.
Sets withZeroContent
Thenextdefinitionwillenableustoestablishtheexistenceof
R
R
f.X/dXincaseswhere
f isboundedontherectangleR,butisnotnecessarilycontinuousforallXinR.
Definition7.1.14
AsubsetEofRhaszerocontentifforeach>0thereisafinite
setofrectanglesT
1
,T
2
,...,T
m
suchthat
E
[m
jD1
T
j
(7.1.24)
and
Xm
jD1
V.T
j
/<:
(7.1.25)
Example7.1.3
Sincetheemptysetiscontainedineveryrectangle,theemptysethas
zerocontent.IfEconsistsoffinitelymanypointsX
1
,X
2
,...,X
m
,thenX
j
canbeenclosed
inarectangleT
j
suchthat
V.T
j
/<
m
; 1j j m:
Then(7.1.24)and(7.1.25)hold,soEhaszerocontent.
Section7.1
DefinitionandExistenceoftheMultipleIntegral
449
Example7.1.4
AnyboundedsetEwithonlyfinitelymanylimitpointshaszerocon-
tent.Toseethis,wefirstobservethatifEhasnolimitpoints,thenitmustbefinite,bythe
Bolzano–Weierstrasstheorem(Theorem1.3.8),andthereforemusthavezerocontent,by
Example7.1.3. NowsupposethatthelimitpointsofEareX
1
,X
2
,...,X
m
. LetR
1
,R
2
,
...,R
m
berectanglessuchthatX
i
2R
0
i
and
V.R
i
/<
2m
; 1im:
(7.1.26)
ThesetofpointsofE thatarenotin[
m
jD1
R
j
has nolimitpoints(why?) and, , being
bounded,mustbefinite(againbytheBolzano–Weierstrasstheorem).Ifthissetcontainsp
points,thenitcanbecoveredbyrectanglesR
0
1
,R
0
2
,...,R
0
p
with
V.R
0
j
/<
2p
; 1j j p:
(7.1.27)
Now,
E
[m
iD1
R
i
!
[
0
@
[p
jD1
R
0
j
1
A
and,from(7.1.26)and(7.1.27),
Xm
iD1
V.R
i
/C
p
X
jD1
V.R
0
j
/<:
Example7.1.5
Iff iscontinuousonŒa;b,thenthecurve
yDf.x/; axb
(7.1.28)
(thatis, theset
˚
.x;y/
ˇ
ˇ
yDf.x/;axb
/, has zerocontentinR2. Tosee e this,
supposethat>0,andchooseı>0suchthat
jf.x/f.x
0
/j< if x;x
0
2Œa;b and jxx
0
j<ı:
(7.1.29)
Thisispossiblebecausef isuniformlycontinuousonŒa;b(Theorem2.2.12).Let
P WaDx
0
<x
1
<<x
n
Db
beapartitionofŒa;bwithkPk<ı,andchoose
1
,
2
,...,
n
sothat
x
i1

i
x
i
; 1i i n:
Then,from(7.1.29),
jf.x/f.
i
/j< if x
i1
xx
i
:
Thismeansthateverypointonthecurve(7.1.28)abovetheintervalŒx
i1
;x
i
isinarect-
anglewitharea2.x
i
x
i1
/(Figure7.1.4). Sincethetotalareaoftheserectanglesis
2.ba/,thecurvehaszerocontent.
450 Chapter7
IntegralsofFunctionsofSeveralVariables
y
x
y = f(ξ
i
) +
y = f(ξ
i
)
y = f(ξ
i
) −
a
b
x
i−1
x
i
ξ
i
Figure7.1.4
ThenextlemmafollowsimmediatelyfromDefinition7.1.14.
Lemma7.1.15
Theunionoffinitelymanysetswithzerocontenthaszerocontent:
Thefollowingtheoremwillenableustodefinemultipleintegralsovermoregeneral
subsetsofR
n
.
Theorem7.1.16
Supposethatfisboundedonarectangle
RDŒa
1
;b
1
Œa
2
;b
2
Œa
n
;b
n
(7.1.30)
andcontinuousexceptonasubsetEofRwithzerocontent:Thenf isintegrableonR:
Proof
Supposethat>0. SinceEhaszerocontent,therearerectanglesT
1
,T
2
,...,
T
m
suchthat
E
[m
jD1
T
j
(7.1.31)
and
Xm
jD1
V.T
j
/<:
(7.1.32)
WemayassumethatT
1
,T
2
,..., T
m
arecontainedinR,since,ifnot,theirintersections
withRwouldbecontainedinR,andstillsatisfy(7.1.31)and(7.1.32).Wemayalsoassume
thatifTisanyrectanglesuchthat
T
\
0
@
[m
jD1
T
0
j
1
A
D;; then T\ED;
(7.1.33)
Section7.1
DefinitionandExistenceoftheMultipleIntegral
451
sinceifthiswerenotso,wecouldmakeitsobyenlargingT
1
,T
2
,...,T
m
slightlywhile
maintaining(7.1.32).Nowsupposethat
T
j
DŒa
1j
;b
1j
Œa
2j
;b
2j
Œa
nj
;b
nj
; 1j j m;
letP
i0
bethepartitionofŒa
i
;b
i
(see(7.1.30))withpartitionpoints
a
i
;b
i
;a
i1
;b
i1
;a
i2
;b
i2
;:::;a
im
;b
im
(thesearenotinincreasingorder),1in,andlet
P
0
DP
10
P
20
P
n0
:
ThenP
0
consistsofrectangleswhoseunionequals[
m
jD1
T
j
andotherrectanglesT
0
1
,T
0
2
,
...,T0
k
thatdonotintersectE.(Weneed(7.1.33)tobesurethatT0
i
\ED;;1ik:/
Ifwelet
BD
[m
jD1
T
j
and C C D
[k
iD1
T
0
i
;
thenRDB[C andf f iscontinuousonthecompactsetC.IfP DfR
1
;R
2
;:::;R
k
gis
arefinementofP
0
,theneverysubrectangleR
j
ofPiscontainedentirelyinBorentirely
inC.Therefore,wecanwrite
S.P/s.P/D†
1
.M
j
m
j
/V.R
j
/C†
2
.M
j
m
j
/V.R
j
/;
(7.1.34)
where†
1
and†
2
aresummationsovervaluesofj forwhichR
j
 BandR
j
 C,
respectively.Nowsupposethat
jf.X/jM
forXinR:
Then
1
.M
j
m
j
/V.R
j
/2M†
1
V.R
j
/D2M
Xm
jD1
V.T
j
/<2M;
(7.1.35)
from(7.1.32). Sincef isuniformlycontinuousonthecompactsetC C (Theorem5.2.14),
thereisaı>0suchthatM
j
m
j
<ifkPk<ıandR
j
C;hence,
2
.M
j
m
j
/V.R
j
/<†
2
V.R
j
/V.R/:
This,(7.1.34),and(7.1.35)implythat
S.P/s.P/<Œ2MCV.R/
ifkPk< ıandP isarefinementofP
0
. Therefore,Theorem7.1.12impliesthatf is
integrableonR.
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