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Section3.2
ExistenceoftheIntegral
133
Proof
If>0,thereisaı>0suchthat
Z
b
a
f.x/dx<s.P/S.P/<
Z
b
a
f.x/dxC
(3.2.18)
ifkPk<ı(Lemma3.2.4).IfisaRiemannsumoff overP,then
s.P/S.P/;
so(3.2.16)and(3.2.18)implythat
L<<LC
ifkPk<ı.NowDefinition3.1.1implies(3.2.17).
Theorems3.2.3and3.2.5implythefollowingtheorem.
Theorem3.2.6
Aboundedfunctionf isintegrableonŒa;bifandonlyif
Z
b
a
f.x/dxD
Z
b
a
f.x/dx:
Thenexttheoremtranslatesthisintoatestthatcanbeconvenientlyapplied.
Theorem3.2.7
Iff isboundedonŒa;b;thenf f isintegrableonŒa;bifandonlyif
foreach>0thereisapartitionPofŒa;bforwhich
S.P/s.P/<:
(3.2.19)
Proof
Weleaveittoyou(Exercise3.2.4)toshowthatif
R
b
a
f.x/dxexists,then(3.2.19)
holdsforkPksufficientlysmall.Thisimpliesthatthestatedconditionisnecessaryforin-
tegrability.Toshowthatitissufficient,weobservethatsince
s.P/
Z
b
a
f.x/dx
Z
b
a
f.x/dxS.P/
forallP,(3.2.19)impliesthat
0
Z
b
a
f.x/dx
Z
b
a
f.x/dx<:
Sincecanbeanypositivenumber,thisimpliesthat
Z
b
a
f.x/dxD
Z
b
a
f.x/dx:
Therefore,
R
b
a
f.x/dxexists,byTheorem3.2.5.
ThenexttwotheoremsareimportantapplicationsofTheorem3.2.7.
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134 Chapter3
IntegralCalculusofFunctionsofOneVariable
Theorem3.2.8
Iff iscontinuousonŒa;b;thenf isintegrableonŒa;b.
Proof
LetPDfx
0
;x
1
;:::;x
n
gbeapartitionofŒa;b.SincefiscontinuousonŒa;b,
therearepointsc
j
andc
0
j
inŒx
j1
;x
j
suchthat
f.c
j
/DM
j
D
sup
x
j1
xx
j
f.x/
and
f.c
0
j
/Dm
j
D
inf
x
j1
xx
j
f.x/
(Theorem2.2.9).Therefore,
S.P/s.P/D
Xn
jD1
f.c
j
/f.c
0
j
/
.x
j
x
j1
/:
(3.2.20)
Sincef isuniformlycontinuousonŒa;b(Theorem2.2.12),thereisforeach>0aı>0
suchthat
jf.x
0
/f.x/j<
ba
ifxandx
0
areinŒa;bandjxx
0
j<ı.IfkPk<ı,thenjc
j
c
0
j
j<ıand,from(3.2.20),
S.P/s.P/<
ba
Xn
jD1
.x
j
x
j1
/D:
Hence,fisintegrableonŒa;b,byTheorem3.2.7.
Theorem3.2.9
Iff ismonotoniconŒa;b;thenf isintegrableonŒa;b.
Proof
LetP Dfx
0
;x
1
;:::;x
n
gbeapartitionofŒa;b.Sincef isnondecreasing,
f.x
j
/DM
j
D
sup
x
j1
xx
j
f.x/
and
f.x
j1
/Dm
j
D
inf
x
j1
xx
j
f.x/:
Hence,
S.P/s.P/D
Xn
jD1
.f.x
j
/f.x
j1
//.x
j
x
j1
/:
Since0<x
j
x
j1
kPkandf.x
j
/f.x
j1
/0,
S.P/s.P/kPk
Xn
jD1
.f.x
j
/f.x
j1
//
DkPk.f.b/f.a//:
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Section3.2
ExistenceoftheIntegral
135
Therefore,
S.P/s.P/< if kPk.f.b/f.a//<;
sof isintegrableonŒa;b,byTheorem3.2.7.
Theprooffornonincreasingf issimilar.
WewillalsouseTheorem3.2.7inthenextsectiontoestablishpropertiesoftheintegral.
InSection3.5wewillstudymoregeneralconditionsforintegrability.
3.2Exercises
1.
CompletetheproofofLemma3.2.1byverifyingEqn.(3.2.3).
2.
Showthatiff isintegrableonŒa;b,then
Z
b
a
f.x/dxD
Z
b
a
f.x/dx:
3.
Prove:Iff isboundedonŒa;b,thereisforeach>0aı>0suchthat
Z
b
a
f.x/dx
Z
b
a
f.x/dx<s.P/
ifkPk<ı.
4.
Prove: Iff isintegrableonŒa;band  > 0,thenS.P/s.P/ < ifkPkis
sufficientlysmall.H
INT
:UseTheorem3.1.4:
5.
Supposethatf isintegrableandgisboundedonŒa;b,andgdiffersfromf f only
atpointsinasetHwiththefollowingproperty:Foreach>0,H canbecovered
byafinitenumberofclosedsubintervalsofŒa;b,thesumofwhoselengthsisless
than.ShowthatgisintegrableonŒa;bandthat
Z
b
a
g.x/dx D
Z
b
a
f.x/dx:
H
INT
:UseExercise3.1.3:
6.
SupposethatgisboundedonŒ˛;ˇ,andletQW˛Dv
0
<v
1
<<v
L
Dˇbe
afixedpartitionofŒ˛;ˇ.Prove:
(a)
Z
ˇ
˛
g.u/duD
L
X
`D1
Z
v
`
v
`1
g.u/duI
(b)
Z
ˇ
˛
g.u/duD
L
X
`D1
Z
v
`
v
`1
g.u/du:
7.
Afunctionf isofboundedvariationonŒa;bifthereisanumberKsuchthat
Xn
jD1
ˇ
ˇ
f.a
j
/f.a
j1
/
ˇ
ˇ
K
wheneveraDa
0
<a
1
<<a
n
Db. (Thesmallestnumberwiththisproperty
isthetotalvariationoff onŒa;b.)
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136 Chapter3
IntegralCalculusofFunctionsofOneVariable
(a)
Prove:IffisofboundedvariationonŒa;b,thenf isboundedonŒa;b.
(b)
Prove: Iff isofboundedvariationonŒa;b,thenf f isintegrableonŒa;b.
H
INT
:UseTheorems3.1.4and3.2.7:
8.
LetP Dfx
0
;x
1
;:::;x
n
gbeapartitionofŒa;b,c
0
Dx
0
Da,c
nC1
Dx
n
Db,
andx
j1
c
j
x
j
,j D1,2,...,n.Verifythat
Xn
jD1
g.c
j
/Œf.x
j
/f.x
j1
/Dg.b/f.b/g.a/f.a/
Xn
jD0
f.x
j
/Œg.c
jC1
/g.c
j
/:
Usethistoprovethatif
R
b
a
f.x/dg.x/exists,thensodoes
R
b
a
g.x/df.x/,and
Z
b
a
g.x/df.x/Df.b/g.b/f.a/g.a/
Z
b
a
f.x/dg.x/:
(ThisistheintegrationbypartsformulaforRiemann–Stieltjesintegrals.)
9.
Letf becontinuousandgbeofboundedvariation(Exercise3.2.7)onŒa;b.
(a)
Showthatif > 0,thereisaı > 0suchthatj 
0
j <=2ifand
0
areRiemann–Stieltjessumsoff withrespecttogoverpartitionsP P andP
0
ofŒa;b,whereP
0
isarefinementofP andkPk k < ı. H
INT
:UseTheo-
rem2.2.12:
(b)
Letıbeaschosenin
(a)
. Supposethat
1
and
2
areRiemann–Stieltjes
sumsoff withrespecttogoveranypartitionsP
1
andP
2
ofŒa;bwithnorm
lessthanı.Showthatj
1

2
j<.
(c)
Ifı>0,letL.ı/bethesupremumofallRiemann–Stieltjessumsoff with
respecttogoverpartitionsofŒa;bwithnormslessthanı.ShowthatL.ı/is
finite.ThenshowthatLDlim
ı!0C
L.ı/exists.H
INT
:UseTheorem2.1.9:
(d)
Showthat
R
b
a
f.x/dg.x/DL.
10.
Showthat
R
b
a
f.x/dg.x/existsiff isofboundedvariationandgiscontinuouson
Œa;b.H
INT
:SeeExercises3.2.8and3.2.9:
3.3PROPERTIESOFTHEINTEGRAL
WenowusetheresultsofSections3.1and3.2toestablishthepropertiesoftheintegral.
Youareprobablyfamiliarwithmostoftheseproperties,butnotwiththeirproofs.
Theorem3.3.1
Iff andgareintegrableonŒa;b;thensoisf Cg;and
Z
b
a
.f Cg/.x/dxD
Z
b
a
f.x/dxC
Z
b
a
g.x/dx:
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Section3.3
PropertiesoftheIntegral
137
Proof
AnyRiemannsumoffCgoverapartitionP Dfx
0
;x
1
;:::;x
n
gofŒa;bcan
bewrittenas
fCg
D
Xn
jD1
Œf.c
j
/Cg.c
j
/.x
j
x
j1
/
D
n
X
jD1
f.c
j
/.x
j
x
j1
/C
n
X
jD1
g.c
j
/.x
j
x
j1
/
D
f
C
g
;
where
f
and
g
areRiemannsumsforf andg. . Definition3.1.1impliesthatif >0
therearepositivenumbersı
1
andı
2
suchthat
ˇ
ˇ
ˇ
ˇ
ˇ
f
Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
<
2
if kPk<ı
1
and
ˇ
ˇ
ˇ
ˇ
ˇ
g
Z
b
a
g.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
<
2
if kPk<ı
2
:
IfkPk<ıDmin.ı
1
2
/,then
ˇ
ˇ
ˇ
ˇ
ˇ
fCg
Z
b
a
f.x/dx
Z
b
a
g.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
D
ˇ
ˇ
ˇ
ˇ
ˇ
f
Z
b
a
f.x/dx
!
C
g
Z
b
a
g.x/dx
!
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
f
Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
C
ˇ
ˇ
ˇ
ˇ
ˇ
g
Z
b
a
g.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
<
2
C
2
D;
sotheconclusionfollowsfromDefinition3.1.1.
ThenexttheoremalsofollowsfromDefinition3.1.1(Exercise3.3.1).
Theorem3.3.2
Iff isintegrableonŒa;bandcisaconstant;thencf f isintegrable
onŒa;band
Z
b
a
cf.x/dxDc
Z
b
a
f.x/dx:
Theorems3.3.1and3.3.2andinductionyieldthefollowingresult(Exercise3.3.2).
Theorem3.3.3
Iff
1
;f
2
;...; f
n
areintegrableonŒa;bandc
1
;c
2
; ...; c
n
are
constants;thenc
1
f
1
Cc
2
f
2
CCc
n
f
n
isintegrableonŒa;band
Z
b
a
.c
1
f
1
Cc
2
f
2
CCc
n
f
n
/.x/dxDc
1
Z
b
a
f
1
.x/dxCc
2
Z
b
a
f
2
.x/dx
CCc
n
Z
b
a
f
n
.x/dx:
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138 Chapter3
IntegralCalculusofFunctionsofOneVariable
Theorem3.3.4
Iff andgareintegrableonŒa;bandf.x/g.x/foraxb;
then
Z
b
a
f.x/dx
Z
b
a
g.x/dx:
(3.3.1)
Proof
Sinceg.x/f.x/0,everylowersumofgfoveranypartitionofŒa;bis
nonnegative.Therefore,
Z
b
a
.g.x/f.x//dx0:
Hence,
Z
b
a
g.x/dx
Z
b
a
f.x/dxD
Z
b
a
.g.x/f.x//dx
D
Z
b
a
.g.x/f.x//dx0;
(3.3.2)
whichyields(3.3.1).(Thefirstequalityin(3.3.2)followsfromTheorems3.3.1and3.3.2;
thesecond,fromTheorem3.2.3.)
Theorem3.3.5
Iff isintegrableonŒa;b;thensoisjfj,and
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
jf.x/jdx:
(3.3.3)
Proof
LetP beapartitionofŒa;banddefine
M
j
Dsup
˚
f.x/
ˇ
ˇ
x
j1
xx
j
;
m
j
Dinf
˚
f.x/
ˇ
ˇ
x
j1
xx
j
;
M
j
Dsup
˚
jf.x/j
ˇ
ˇ
x
j1
xx
j
;
m
j
Dinf
˚
jf.x/j
ˇ
ˇ
x
j1
xx
j
:
Then
M
j
m
j
Dsup
˚
jf.x/jjf.x
0
/j
ˇ
ˇ
x
j1
x;x
0
x
j
sup
˚
jf.x/f.x
0
/j
ˇ
ˇ
x
j1
x;x
0
x
j
DM
j
m
j
:
(3.3.4)
Therefore,
S.P/
s.P/S.P/s.P/;
wheretheupperandlowersumsontheleftareassociatedwithjfjandthoseontherightare
associatedwithf.Nowsupposethat>0.Sincef isintegrableonŒa;b,Theorem3.2.7
impliesthatthereisapartitionP ofŒa;bsuchthatS.P/s.P/ / <. Thisinequality
and(3.3.4)implythat
S.P/
s.P/<. Therefore,jfjisintegrableonŒa;b,againby
Theorem3.2.7.
Since
f.x/jf.x/j and
f.x/jf.x/j; axb;
Section3.3
PropertiesoftheIntegral
139
Theorems3.3.2and3.3.4implythat
Z
b
a
f.x/dx
Z
b
a
jf.x/jdx and d 
Z
b
a
f.x/dx
Z
b
a
jf.x/jdx;
whichimplies(3.3.3).
Theorem3.3.6
Iff andgareintegrableonŒa;b;thensoistheproductfg:
Proof
Weconsiderthecasewheref andgarenonnegative,andleavetherestofthe
prooftoyou(Exercise3.3.4). Thesubscriptsf,g, , andfginthefollowingargument
identifythefunctionswithwhichthevariousquantitiesareassociated. Weassumethat
neitherf norgisidenticallyzeroonŒa;b,sincetheconclusionisobviousifoneofthem
is.
IfPDfx
0
;x
1
;:::;x
n
gisapartitionofŒa;b,then
S
fg
.P/s
fg
.p/D
n
X
jD1
.M
fg;j
m
fg;j
/.x
j
x
j1
/:
(3.3.5)
Sincef andgarenonnegative,M
fg;j
M
f;j
M
g;j
andm
fg;j
m
f;j
m
g;j
.Hence,
M
fg;j
m
fg;j
M
f;j
M
g;j
m
f;j
m
g;j
D.M
f;j
m
f;j
/M
g;j
Cm
f;j
.M
g;j
m
g;j
/
M
g
.M
f;j
m
f;j
/CM
f
.M
g;j
m
g;j
/;
whereM
f
andM
g
areupperboundsforf andgonŒa;b. From(3.3.5)andthelast
inequality,
S
fg
.P/s
fg
.P/M
g
ŒS
f
.P/s
f
.P/CM
f
ŒS
g
.P/s
g
.P/:
(3.3.6)
Nowsupposethat > 0. Theorem3.2.7impliesthattherearepartitionsP
1
andP
2
of
Œa;bsuchthat
S
f
.P
1
/s
f
.P
1
/<
2M
g
and S
g
.P
2
/s
g
.P
2
/<
2M
f
:
(3.3.7)
IfP isarefinementofbothP
1
andP
2
,then(3.3.7)andLemma3.2.1implythat
S
f
.P/s
f
.P/<
2M
g
and S
g
.P/s
g
.P/<
2M
f
:
Thisand(3.3.6)yield
S
fg
.P/s
fg
.P/<
2
C
2
D:
Therefore,fgisintegrableonŒa;b,byTheorem3.2.7.
140 Chapter3
IntegralCalculusofFunctionsofOneVariable
Theorem3.3.7(FirstMeanValueTheoremforIntegrals)
Supposethat
uiscontinuousandvisintegrableandnonnegativeonŒa;b:Then
Z
b
a
u.x/v.x/dxDu.c/
Z
b
a
v.x/dx
(3.3.8)
forsomecinŒa;b.
Proof
FromTheorem3.2.8,uisintegrableonŒa;b.Therefore,Theorem3.3.6implies
thattheintegralontheleftexists.IfmDmin
˚
u.x/
ˇ
ˇ
axb
andMDmax
˚
u.x/
ˇ
ˇ
axb
(recallTheorem2.2.9),then
mu.x/M
and,sincev.x/0,
mv.x/u.x/v.x/Mv.x/:
Therefore,Theorems3.3.2and3.3.4implythat
m
Z
b
a
v.x/dx
Z
b
a
u.x/v.x/dxM
Z
b
a
v.x/dx:
(3.3.9)
Thisimpliesthat(3.3.8)holdsforanycinŒa;bif
R
b
a
v.x/dxD0.If
R
b
a
v.x/dx¤0,
let
uD
Z
b
a
u.x/v.x/dx
Z
b
a
v.x/dx
(3.3.10)
Since
R
b
a
v.x/dx > 0inthiscase(why?),(3.3.9)impliesthatm 
u  M, , andthe
intermediatevaluetheorem(Theorem2.2.10)impliesthat
uDu.c/forsomecinŒa;b.
Thisimplies(3.3.8).
Ifv.x/1,then(3.3.10)reducesto
uD
1
ba
Z
b
a
u.x/dx;
so
uistheaverageofu.x/overŒa;b.Moregenerally,ifvisanynonnegativeintegrable
functionsuchthat
R
b
a
v.x/dx¤0,then
uin(3.3.10)istheweightedaverageofu.x/over
Œa;bwithrespecttov. Theorem3.3.7saysthatacontinuousfunctionassumesanysuch
weightedaverageatsomepointinŒa;b.
Theorem3.3.8
IffisintegrableonŒa;bandaa
1
<b
1
b;thenfisintegrable
onŒa
1
;b
1
:
Section3.3
PropertiesoftheIntegral
141
Proof
Supposethat>0.FromTheorem3.2.7,thereisapartitionP Dfx
0
;x
1
;:::;x
n
g
ofŒa;bsuchthat
S.P/s.P/D
Xn
jD1
.M
j
m
j
/.x
j
x
j1
/<:
(3.3.11)
Wemayassumethata
1
andb
1
arepartitionpointsofP,becauseifnottheycanbeinserted
toobtainarefinementPsuchthatS.P0/s.P0/ S.P/s.P/(Lemma3.2.1). . Let
a
1
Dx
r
andb
1
Dx
s
.Sinceeverytermin(3.3.11)isnonnegative,
Xs
jDrC1
.M
j
m
j
/.x
j
x
j1
/<:
Thus,
P D D fx
r
;x
rC1
;:::;x
s
gisapartitionofŒa
1
;b
1
overwhichtheupperandlower
sumsoff satisfy
S.
P/s.
P/<:
Therefore,f isintegrableonŒa
1
;b
1
,byTheorem3.2.7.
Weleavetheproofofthenexttheoremtoyou(Exercise3.3.8).
Theorem3.3.9
Iff isintegrableonŒa;bandŒb;c;thenf isintegrableonŒa;c;
and
Z
c
a
f.x/dxD
Z
b
a
f.x/dxC
Z
c
b
f.x/dx:
(3.3.12)
Sofarwehavedefined
R
ˇ
˛
f.x/dxonlyforthecasewhere˛<ˇ.Nowwedefine
Z
˛
ˇ
f.x/dxD
Z
ˇ
˛
f.x/dx
if˛<ˇ,and
Z
˛
˛
f.x/dxD0:
Withtheseconventions,(3.3.12)holdsnomatterwhattherelativeorderofa, b,andc,
providedthatfisintegrableonsomeclosedintervalcontainingthem(Exercise3.3.9).
Theorem3.3.8andthesedefinitionsenableustodefineafunctionF.x/D
R
x
c
f.t/dt,
wherecisanarbitrary,butfixed,pointinŒa;b.
Theorem3.3.10
Iff isintegrableonŒa;banda a   c   b;thenthefunctionF
definedby
F.x/D
Z
x
c
f.t/dt
satisfiesaLipschitzconditiononŒa;b;andisthereforecontinuousonŒa;b:
142 Chapter3
IntegralCalculusofFunctionsofOneVariable
Proof
Ifxandx
0
areinŒa;b,then
F.x/F.x
0
/D
Z
x
c
f.t/dt
Z
x
0
c
f.t/dtD
Z
x
x
0
f.t/dt;
byTheorem3.3.9andtheconventionsjustadopted. Sincejf.t/j  K.a t b/for
someconstantK,
ˇ
ˇ
ˇ
ˇ
Z
x
x0
f.t/dt
ˇ
ˇ
ˇ
ˇ
Kjxx
0
j; ax;x
0
b
(Theorem3.3.5),so
jF.x/F.x
0
/jKjxx
0
j; ax;x
0
b:
Theorem3.3.11
Iff isintegrableonŒa;bandacb;thenF.x/D
R
x
c
f.t/dt
isdifferentiableatanypointx
0
in.a;b/wheref iscontinuous;withF
0
.x
0
/Df.x
0
/:If
f iscontinuousfromtherightata;thenF
0
C
.a/Df.a/. Iff f iscontinuousfromtheleft
atb;thenF
0
.b/Df.b/:
Proof
Weconsiderthecase where a < x
0
< b b andleave theresttoyou(Exer-
cise3.3.14).Since
1
xx
0
Z
x
x
0
f.x
0
/dtDf.x
0
/;
wecanwrite
F.x/F.x
0
/
xx
0
f.x
0
/D
1
xx
0
Z
x
x
0
Œf.t/f.x
0
/dt:
FromthisandTheorem3.3.5,
ˇ
ˇ
ˇ
ˇ
F.x/F.x
0
/
xx
0
f.x
0
/
ˇ
ˇ
ˇ
ˇ
1
jxx
0
j
ˇ
ˇ
ˇ
ˇ
Z
x
x
0
jf.t/f.x
0
/jdt
ˇ
ˇ
ˇ
ˇ
:
(3.3.13)
(Whydoweneedtheabsolutevaluebarsoutsidetheintegral?) Sincef iscontinuousat
x
0
,thereisforeach>0aı>0suchthat
jf.t/f.x
0
/j< if jxx
0
j<ı
andtisbetweenxandx
0
.Therefore,from(3.3.13),
ˇ
ˇ
ˇ
ˇ
F.x/F.x
0
/
xx
0
f.x
0
/
ˇ
ˇ
ˇ
ˇ
<
jxx
0
j
jxx
0
j
D if 0<jxx
0
j<ı:
Hence,F
0
.x
0
/Df.x
0
/.
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