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Section3.3
PropertiesoftheIntegral
143
Example3.3.1
If
f.x/D
(
x;
0x1;
xC1; 1<x2;
thenthefunction
F.x/D
Z
x
0
f.t/dtD
8
ˆ
ˆ
<
ˆ
ˆ
:
x
2
2
;
0<x1;
x
2
2
Cx1; 1<x2;
iscontinuousonŒ0;2.AsimpliedbyTheorem3.3.11,
F
0
.x/D
8
<
:
xDf.x/;
0<x<1;
xC1Df.x/; 1<x<2;
F
0
C
.0/D lim
x!0C
F.x/F.0/
x
D lim
x!0C
.x
2
=2/0
x
D0Df.0/;
F
0
.2/D lim
x!2
F.x/F.2/
x2
D lim
x!2
.x
2
=2/Cx13
x2
D lim
x!2
xC4
2
D3Df.2/:
F
0
.1/D1 and F
0
C
.1/D2:
Thenexttheoremrelatesintegrationanddifferentiationinanotherway.
Theorem3.3.12
SupposethatF iscontinuousontheclosedintervalŒa;banddif-
ferentiableontheopeninterval.a;b/;andf isintegrableonŒa;b:Supposealsothat
F
0
.x/Df.x/; a<x<b:
Then
Z
b
a
f.x/dxDF.b/F.a/:
(3.3.14)
Proof
IfP Dfx
0
;x
1
;:::;x
n
gisapartitionofŒa;b,then
F.b/F.a/D
n
X
jD1
.F.x
j
/F.x
j1
//:
(3.3.15)
FromTheorem2.3.11,thereisineachopeninterval.x
j1
;x
j
/apointc
j
suchthat
F.x
j
/F.x
j1
/Df.c
j
/.x
j
x
j1
/:
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144 Chapter3
IntegralCalculusofFunctionsofOneVariable
Hence,(3.3.15)canbewrittenas
F.b/F.a/D
Xn
jD1
f.c
j
/.x
j
x
j1
/D;
whereisaRiemannsumforf overP.Sincef isintegrableonŒa;b,thereisforeach
>0aı>0suchthat
ˇ
ˇ
ˇ
ˇ
ˇ

Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
< if kPk<ı:
Therefore,
ˇ
ˇ
ˇ
ˇ
ˇ
F.b/F.a/
Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
<
forevery>0,whichimplies(3.3.14).
Corollary3.3.13
Iff
0
isintegrableonŒa;b;then
Z
b
a
f
0
.x/dxDf.b/f.a/:
Proof
ApplyTheorem3.3.12withF andf replacedbyf andf
0
,respectively.
AfunctionFisanantiderivativeoff onŒa;bifF iscontinuousonŒa;banddiffer-
entiableon.a;b/,with
F
0
.x/Df.x/; a<x<b:
IfF isanantiderivativeoff onŒa;b,thensoisF F Ccforanyconstantc. Conversely,
ifF
1
andF
2
areantiderivativesoff onŒa;b,thenF
1
F
2
isconstantonŒa;b(Theo-
rem2.3.12).Theorem3.3.12showsthatantiderivativescanbeusedtoevaluateintegrals.
Theorem3.3.14(FundamentalTheoremofCalculus)
Iff iscontinu-
ousonŒa;b;thenf hasanantiderivativeonŒa;b:Moreover;ifF isanyantiderivative
off onŒa;b;then
Z
b
a
f.x/dxDF.b/F.a/:
Proof
ThefunctionF
0
.x/ D
R
x
a
f.t/dt iscontinuousonŒa;bbyTheorem 3.3.10,
andF
0
0
.x/Df.x/on.a;b/byTheorem3.3.11. Therefore,F
0
isanantiderivativeoff
onŒa;b.NowletF DF
0
Cc(cDconstant)beanarbitraryantiderivativeoff onŒa;b.
Then
F.b/F.a/D
Z
b
a
f.x/dxCc
Z
a
a
f.x/dxcD
Z
b
a
f.x/dx:
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Section3.3
PropertiesoftheIntegral
145
Whenapplyingthistheorem,wewillusethefamiliarnotation
F.b/F.a/DF.x/
ˇ
ˇ
ˇ
ˇ
b
a
:
Theorem3.3.15(IntegrationbyParts)
Ifu
0
andv
0
areintegrableonŒa;b;
then
Z
b
a
u.x/v
0
.x/dxDu.x/v.x/
ˇ
ˇ
ˇ
ˇ
b
a
Z
b
a
v.x/u
0
.x/dx:
(3.3.16)
Proof
SinceuandvarecontinuousonŒa;b(Theorem2.3.3),theyareintegrableon
Œa;b.Therefore,Theorems3.3.1and3.3.6implythatthefunction
.uv/
0
Du
0
vCuv
0
isintegrableonŒa;b,andTheorem3.3.12impliesthat
Z
b
a
Œu.x/v
0
.x/Cu
0
.x/v.x/dxDu.x/v.x/
ˇ
ˇ
ˇ
ˇ
b
a
;
whichimplies(3.3.16).
WewilluseTheorem3.3.15hereandinthenextsectiontoobtainotherresults.
Theorem3.3.16(Second MeanValueTheoremforIntegrals)
Suppose
thatf
0
isnonnegativeandintegrableandgiscontinuousonŒa;b:Then
Z
b
a
f.x/g.x/dxDf.a/
Z
c
a
g.x/dxCf.b/
Z
b
c
g.x/dx
(3.3.17)
forsomecinŒa;b:
Proof
Sincef isdifferentiableonŒa;b,itiscontinuousonŒa;b(Theorem 2.3.3).
SincegiscontinuousonŒa;b,soisfg(Theorem2.2.5).Therefore,Theorem3.2.8implies
thattheintegralsin(3.3.17)exist.If
G.x/D
Z
x
a
g.t/dt;
(3.3.18)
thenG
0
.x/Dg.x/;a<x<b(Theorem3.3.11).Therefore,Theorem3.3.15withuDf
andvDGyields
Z
b
a
f.x/g.x/dxDf.x/G.x/
ˇ
ˇ
ˇ
ˇ
b
a
Z
b
a
f
0
.x/G.x/dx:
(3.3.19)
Sincef0isnonnegativeandGiscontinuous,Theorem3.3.7impliesthat
Z
b
a
f
0
.x/G.x/dxDG.c/
Z
b
a
f
0
.x/dx
(3.3.20)
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146 Chapter3
IntegralCalculusofFunctionsofOneVariable
forsomecinŒa;b.FromCorollary3.3.12,
Z
b
a
f
0
.x/dxDf.b/f.a/:
Fromthisand(3.3.18),(3.3.20)canberewrittenas
Z
b
a
f
0
.x/G.x/dxD.f.b/f.a//
Z
c
a
g.x/dx:
Substitutingthisinto(3.3.19)andnotingthatG.a/D0yields
Z
b
a
f.x/g.x/dxDf.b/
Z
b
a
g.x/dx.f.b/f.a//
Z
c
a
g.x/dx;
Df.a/
Z
c
a
g.x/dxCf.b/
Z
b
a
g.x/dx
Z
a
c
g.x/dx
!
Df.a/
Z
c
a
g.x/dxCf.b/
Z
b
c
g.x/dx:
Change ofVariable
Thefollowingtheoremonchangeofvariableisusefulforevaluatingintegrals.
Theorem3.3.17
Supposethatthetransformationx D .t/mapstheintervalc 
tdintotheintervalaxb;with.c/D˛and.d/Dˇ;andletf becontinuous
onŒa;b:LetbeintegrableonŒc;d:Then
Z
ˇ
˛
f.x/dxD
Z
d
c
f..t//
0
.t/dt:
(3.3.21)
Proof
Bothintegralsin(3.3.21)exist:theoneontheleftbyTheorem3.2.8,theoneon
therightbyTheorems3.2.8and3.3.6andthecontinuityoff..t//. ByTheorem3.3.11,
thefunction
F.x/D
Z
x
a
f.y/dy
isanantiderivativeoff onŒa;band,therefore,alsoontheclosedintervalwithendpoints
˛andˇ.Hence,byTheorem3.3.14,
Z
ˇ
˛
f.x/dxDF.ˇ/F.˛/:
(3.3.22)
Bythechainrule,thefunction
G.t/DF..t//
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Section3.3
PropertiesoftheIntegral
147
isanantiderivativeoff..t//
0
.t/onŒc;d,andTheorem3.3.12impliesthat
Z
d
c
f..t//
0
.t/dtDG.d/G.c/DF..d//F..c//
DF.ˇ/F.˛/:
Comparingthiswith(3.3.22)yields(3.3.21).
Example3.3.2
Toevaluatetheintegral
ID
Z
1=
p
2
1=
p
2
.12x
2
/.1x
2
/
1=2
dx
welet
f.x/D.12x
2
/.1x
2
/
1=2
; 1=
p
2x1=
p
2;
and
xD.t/Dsint; =4t=4:
Then
0
.t/Dcostand
I D
Z
1=
p
2
1=
p
2
f.x/dxD
Z
=4
=4
f.sint/costdt
D
Z
=4
=4
.12sin
2
t/.1sin
2
t/
1=2
costdt:
(3.3.23)
.1sin
2
t/
1=2
Dcost;=4t=4
and
12sin
2
tDcos2t;
(3.3.23)yields
ID
Z
=4
=4
cos2tdtD
sin2t
2
ˇ
ˇ
ˇ
ˇ
=4
=4
D1:
Example3.3.3
Toevaluatetheintegral
ID
Z
5
0
sint
2Ccost
dt;
wetake.t/Dcost.Then
0
.t/Dsintand
I D
Z
5
0
0
.t/
2C.t/
dtD
Z
5
0
f..t//
0
.t/dt;
where
f.x/D
1
2Cx
:
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148 Chapter3
IntegralCalculusofFunctionsofOneVariable
Therefore,since.0/D1and.5/D1,
I D
Z
1
1
dx
2Cx
Dlog.2Cx/
ˇ
ˇ
ˇ
ˇ
1
1
Dlog3:
TheseexamplesillustratetwowaystouseTheorem3.3.17.InExample3.3.2weevalu-
atedtheleftsideof(3.3.21)bytransformingittotherightsidewithasuitablesubstitution
x D.t/,whileinExample3.3.3weevaluatedtherightsideof(3.3.21)byrecognizing
thatitcouldbeobtainedfromtheleftsidebyasuitablesubstitution.
Thefollowingtheoremshowsthattheruleforchangeofvariableremainsvalidunder
weakerassumptionsonf ifismonotonic.
Theorem3.3.18
Supposethat
0
isintegrableandismonotoniconŒc;d;andthe
transformationxD.t/mapsŒc;dontoŒa;b:Letf beboundedonŒa;b:Then
g.t/Df..t//
0
.t/
isintegrableonŒc;difandonlyiff isintegrableoverŒa;b;andinthiscase
Z
b
a
f.x/dxD
Z
d
c
f..t//j
0
.t/jdt:
Proof
Weconsiderthecasewheref isnonnegativeandisnondecreasing,andleave
thetherestoftheprooftoyou(Exercises3.3.20and3.3.21).
Firstassumethatisincreasing.Weshowﬁrstthat
Z
b
a
f.x/dxD
Z
d
c
f..t//
0
.t/dt:
(3.3.24)
Let
P Dft
0
;t
1
;:::;t
n
gbeapartitionofŒc;dandP Dfx
0
;x
1
;:::;x
n
gwithx
j
D.t
j
/
bethecorrespondingpartitionofŒa;b.Deﬁne
U
j
Dsup
˚
0
.t/
ˇ
ˇ
t
j1
tt
j
;
u
j
Dinf
˚
0
.t/
ˇ
ˇ
t
j1
tt
j
;
M
j
Dsup
˚
f.x/
ˇ
ˇ
x
j1
xx
j
;
and
M
j
Dsup
˚
f..t//
0
.t/
ˇ
ˇ
t
j1
tt
j
:
Sinceisincreasing,u
j
0.Therefore,
0u
j

0
.t/U
j
; t
j1
tt
j
:
Sincef isnonnegative,thisimpliesthat
0f..t//u
j
f..t//
0
.t/f..t//U
j
; t
j1
tt
j
:
Therefore,
M
j
u
j
M
j
M
j
U
j
;
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Section3.3
PropertiesoftheIntegral
149
whichimpliesthat
M
j
DM
j
j
;
(3.3.25)
where
u
j

j
U
j
:
(3.3.26)
Nowconsidertheuppersums
S.
P/D
Xn
jD1
M
j
.t
j
t
j1
/ and S.P/D
Xn
jD1
M
j
.x
j
x
j1
/:
(3.3.27)
Fromthemeanvaluetheorem,
x
j
x
j1
D.t
j
/.t
j1
/D
0
.
j
/.t
j
t
j1
/;
(3.3.28)
wheret
j1
<
j
<t
j
,so
u
j

0
.
j
/U
j
:
(3.3.29)
From(3.3.25),(3.3.27),and(3.3.28),
S.
P/S.P/D
Xn
jD1
M
j
.
j

0
.
j
//.t
j
t
j1
/:
(3.3.30)
Nowsupposethatjf.x/jM,axb.Then(3.3.26),(3.3.29),and(3.3.30)imply
that
ˇ
ˇ
S.
P/S.P/
ˇ
ˇ
M
n
X
jD1
.U
j
u
j
/.t
j
t
j1
/:
Thesumontherightisthedifferencebetweentheupperandlowersumsofover
P.
Pk
sufﬁcientlysmall(Exercise3.2.4).
From(3.3.28),kPkKk
Pkifj
0
.t/jK,ctd. Hence,Lemma3.2.4implies
that
ˇ
ˇ
ˇ
ˇ
ˇ
S.P/
Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
<
3
and
ˇ
ˇ
ˇ
ˇ
ˇ
S.
P/
Z
d
c
f..t//
0
.t/dt
ˇ
ˇ
ˇ
ˇ
ˇ
<
3
(3.3.31)
ifk
Pkissufﬁcientlysmall.Now
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
f.x/dx
Z
d
c
f..t//
0
.t/dt
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
f.x/dxS.P/
ˇ
ˇ
ˇ
ˇ
ˇ
CjS.P/
S.
P/j
C
ˇ
ˇ
ˇ
ˇ
ˇ
S.
P/
Z
d
c
f..t//
0
.t/dt
ˇ
ˇ
ˇ
ˇ
ˇ
:
Choosing
P sothatjS.P/
S.
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
f.x/dx
Z
d
c
f..t//
0
.t/dt
ˇ
ˇ
ˇ
ˇ
ˇ
<:
Sinceisanarbitrarypositivenumber,thisimplies(3.3.24).
150 Chapter3
IntegralCalculusofFunctionsofOneVariable
Ifisnondecreasing(ratherthanincreasing),itmayhappenthatx
j1
Dx
j
forsome
valuesofj;however,thisisnorealcomplication,sinceitsimplymeansthatsometermsin
S.P/vanish.
Byapplying(3.3.24)tof,weinferthat
Z
b
a
f.x/dxD
Z
d
c
f..t//
0
.t/dt;
(3.3.32)
since
Z
b
a
.f/.x/dxD
Z
b
a
f.x/dx
and
Z
d
c
.f..t/
0
.t//dtD
Z
d
c
f..t//
0
.t/dt:
Nowsupposethatf isintegrableonŒa;b.Then
Z
b
a
f.x/dxD
Z
b
a
f.x/dxD
Z
b
a
f.x/dx;
byTheorem3.2.3.Fromthis,(3.3.24),and(3.3.32),
Z
d
c
f..t//
0
.t/dt D
Z
d
c
f..t//
0
.t/dtD
Z
b
a
f.x/dx:
ThisandTheorem3.2.5(appliedtof..t//
0
.t/)implythatf..t//
0
.t/isintegrableon
Œc;dand
Z
b
a
f.x/dxD
Z
d
c
f..t//
0
.t/dt:
(3.3.33)
Asimilarargumentshowsthatiff..t//
0
.t/isintegrableonŒc;d,thenf isintegrable
onŒa;b,and(3.3.33)holds.
3.3Exercises
1.
ProveTheorem3.3.2.
2.
ProveTheorem3.3.3.
3.
CanjfjbeintegrableonŒa;biff isnot?
4.
CompletetheproofofTheorem3.3.6.H
INT
:Thepartialproofgivenaboveimplies
thatifm
1
andm
2
arelowerboundsforf andgrespectivelyonŒa;b;then
.f m
1
/.gm
2
/isintegrableonŒa;b:
5.
Prove:Iff isintegrableonŒa;bandjf.x/j>0foraxb,then1=f is
integrableonŒa;b
Section3.3
PropertiesoftheIntegral
151
6.
Supposethatf isintegrableonŒa;banddeﬁne
f
C
.x/D
(
f.x/
iff.x/0;
0
iff.x/<0,
and f
.x/D
(
0
iff.x/0;
f.x/
iff.x/<0.
ShowthatfandfareintegrableonŒa;b,and
Z
b
a
f.x/dxD
Z
b
a
f
C
.x/dxC
Z
b
a
f
.x/dx:
7.
Findtheweightedaverage
uofu.x/overŒa;bwithrespecttov,andﬁndapointc
inŒa;bsuchthatu.c/D
u.
(a)
u.x/Dx,
v.x/Dx,
Œa;bDŒ0;1
(b)
u.x/Dsinx, v.x/Dx
2
, Œa;bDŒ1;1
(c)
u.x/Dx
2
,
v.x/De
x
, Œa;bDŒ0;1
8.
ProveTheorem3.3.9.
9.
Showthat
Z
c
a
f.x/dxD
Z
b
a
f.x/dxC
Z
c
b
f.x/dx
forallpossiblerelativeorderingsofa,b,andc,providedthatf isintegrableona
closedintervalcontainingthem.
10.
0
<a
1
<<a
n
Db,then
Z
b
a
f.x/dxD
Z
a
1
a
0
f.x/dxC
Z
a
2
a
1
f.x/dxCC
Z
a
n
a
n1
f.x/dx:
11.
Supposethatf iscontinuousonŒa;bandP Dfx
0
;x
1
;:::;x
n
gisapartitionof
Œa;b.ShowthatthereisaRiemannsumoffoverPthatequals
R
b
a
f.x/dx.
12.
Supposethatf
0
existsandjf
0
.x/jMonŒa;b.ShowthatanyRiemannsum
off overanypartitionPofŒa;bsatisﬁes
ˇ
ˇ
ˇ
ˇ
ˇ

Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
M.ba/kPk:
H
INT
:SeeExercise3.3.11:
13.
Prove:Iff isintegrableandf.x/0onŒa;b,then
R
b
a
f.x/dx0,withstrict
inequalityiff iscontinuousandpositiveatsomepointinŒa;b.
14.
CompletetheproofofTheorem3.3.11.
15.
StatetheoremsanalogoustoTheorems3.3.10and3.3.11forthefunction
G.x/D
Z
c
x
f.t/dt;
andshowhowyourtheoremscanbeobtainedfromthem.