c# pdf free : Add links to pdf application control cloud windows web page azure class TRENCH_REAL_ANALYSIS13-part230

Section3.1
DefinitionoftheIntegral
123
Considertheuppersum
S.P
0
/D
Xn
jD1
M
j
.x
j
x
j1
/:
Thereareatmostkvaluesofj inthissumforwhichM
j
=2,andM
j
1evenfor
these. Thecontributionofthesetermstothesumislessthank.=2k/D=2,becauseof
(3.1.10).SinceM
j
<=2forallothervaluesofj,thesumoftheothertermsislessthan
2
n
X
jD1
.x
j
x
j1
/D
2
.x
n
x
0
/D
2
.21/D
2
:
Therefore,S.P
0
/<and,sincecanbechosenassmallaswewish,nopositivenumber
islessthanalluppersums.Thisproves(3.1.9).
ThemotivationforDefinition3.1.3canbeseenbyagainconsideringtheideaofarea
underacurve. Figure3.1.5showsthegraphofapositivefunctionyDf.x/,axb,
withŒa;bpartitionedintofoursubintervals.
a
x
1
x
2
x
3
b
y = f(x)
y
x
Figure3.1.5
Theupperandlowersumsoffoverthispartitioncanbeinterpretedasthesumsoftheareas
oftherectanglessurmountedbythesolidanddashedlines,respectively.Thisindicatesthat
asensibledefinitionofareaAunderthecurvemustadmittheinequalities
s.P/AS.P/
foreverypartitionP ofŒa;b.Thus,Amustbeanupperboundforalllowersumsanda
lowerboundforalluppersumsoff overpartitionsofŒa;b.If
Z
b
a
f.x/dxD
Z
b
a
f.x/dx;
(3.1.11)
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124 Chapter3
IntegralCalculusofFunctionsofOneVariable
thereisonlyonenumber, thecommonvalueoftheupperandlowerintegrals,withthis
property,andwedefineAtobethatnumber;if(3.1.11)doesnothold,thenAisnotdefined.
Wewillseebelowthatthisdefinitionofareaisconsistentwiththedefinitionstatedearlier
intermsofRiemannsums.
Example3.1.7
ReturningtoExample3.1.3,considerthefunction
f.x/Dx; 1x2:
IfP Dfx
0
;x
1
;:::;x
n
gisapartitionofŒ1;2,then,sincef isincreasing,
M
j
Df.x
j
/Dx
j
and m
j
Df.x
j1
/Dx
j1
:
Hence,
S.P/D
Xn
jD1
x
j
.x
j
x
j1
/
(3.1.12)
and
s.P/D
Xn
jD1
x
j1
.x
j
x
j1
/:
(3.1.13)
Bywriting
x
j
D
x
j
Cx
j1
2
C
x
j
x
j1
2
;
weseefrom(3.1.12)that
S.P/D
1
2
Xn
jD1
.x
2
j
x
2
j1
/C
1
2
Xn
jD1
.x
j
x
j1
/
2
D
1
2
.2
2
1
2
/C
1
2
Xn
jD1
.x
j
x
j1
/
2
:
(3.1.14)
Since
0<
Xn
jD1
.x
j
x
j1
/
2
kPk
Xn
jD1
.x
j
x
j1
/DkPk.21/;
(3.1.14)impliesthat
3
2
<S.P/
3
2
C
kPk
2
:
SincekPkcanbemadeassmallasweplease,Definition3.1.3impliesthat
Z
b
a
f.x/dxD
3
2
:
Asimilarargumentstartingfrom(3.1.13)showsthat
3
2
kPk
2
s.P/<
3
2
;
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Section3.1
DefinitionoftheIntegral
125
so
Z
b
a
f.x/dxD
3
2
:
Sincetheupperandlowerintegralsbothequal3=2,theareaunderthecurveis3=2accord-
ingtoournewdefinition.ThisisconsistentwiththeresultinExample3.1.3.
TheRiemann–StieltjesIntegral
TheRiemann–StieltjesintegralisanimportantgeneralizationoftheRiemannintegral.We
defineithere,butconfineourstudyofittotheexercisesinthisandothersectionsofthis
chapter.
Definition3.1.5
Letf andgbedefinedonŒa;b.Wesaythatf isRiemann–Stieltjes
integrablewithrespecttogonŒa;bifthereisanumberLwiththefollowingproperty:
Forevery>0,thereisaı>0suchthat
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
Xn
jD1
f.c
j
/
g.x
j
/g.x
j1
/
L
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
<;
(3.1.15)
providedonlythatP Dfx
0
;x
1
;:::;x
n
gisapartitionofŒa;bsuchthatkPk<ıand
x
j1
c
j
x
j
; j j D1;2;:::;n:
Inthiscase, wesaythatListheRiemann–Stieltjesintegraloff withrespecttogover
Œa;b,andwrite
Z
b
a
f.x/dg.x/DL:
Thesum
Xn
jD1
f.c
j
/
g.x
j
/g.x
j1
/
in(3.1.15)isaRiemann–Stieltjessumoff withrespecttogoverthepartitionP.
3.1Exercises
1.
ShowthattherecannotbemorethanonenumberLthatsatisfiesDefinition3.1.1.
2. (a)
Prove:If
R
b
a
f.x/dxexists,thenforevery>0,thereisaı>0suchthat
j
1

2
j<if
1
and
2
areRiemannsumsoff overpartitionsP
1
andP
2
ofŒa;bwithnormslessthanı.
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126 Chapter3
IntegralCalculusofFunctionsofOneVariable
(b)
SupposethatthereisanM>0suchthat,foreveryı>0,thereareRiemann
sums
1
and
2
overapartitionPofŒa;bwithkPk<ısuchthatj
1

2
j
M.Use
(a)
toprovethatf isnotintegrableoverŒa;b.
3.
Supposethat
R
b
a
f.x/dxexistsandthereisanumberAsuchthat,forevery>0
andı>0,thereisapartitionP ofŒa;bwithkPk<ıandaRiemannsumoff
overPthatsatisfiestheinequalityjAj<.Showthat
R
b
a
f.x/dxDA.
4.
ProvedirectlyfromDefinition3.1.1that
Z
b
a
x
2
dxD
b3a3
3
:
Donotassumeinadvancethattheintegralexists. Theproofofthisispartofthe
problem. H
INT
:LetP Dfx
0
;x
2
;:::;x
n
gbeanarbitrarypartitionofŒa;b:Use
themeanvaluetheoremtoshowthat
b3a3
3
D
Xn
jD1
d
2
j
.x
j
x
j1
/
forsomepointsd
1
;..., d
n
;wherex
j1
< d
j
< x
j
. Thenrelatethissumto
arbitraryRiemannsumsforf.x/Dx
2
overP:
5.
GeneralizetheproofofExercise3.1.4toshowdirectlyfromDefinition3.1.1that
Z
b
a
x
m
dxD
b
mC1
a
mC1
mC1
ifmisaninteger0.
6.
ProvedirectlyfromDefinition3.1.1thatf.x/isintegrableonŒa;bifandonlyif
f.x/isintegrableonŒb;a,and,inthiscase,
Z
b
a
f.x/dxD
Z
a
b
f.x/dx:
7.
Letf beboundedonŒa;bandletP P beapartitionofŒa;b.Prove:Thelowersum
s.P/off overP P istheinfimumofthesetofallRiemannsumsoff f overP.
8.
Letf bedefinedonŒa;bandletPDfx
0
;x
1
;:::;x
n
gbeapartitionofŒa;b.
(a)
Prove:Iff iscontinuousonŒa;b,thens.P/andS.P/areRiemannsumsof
foverP.
(b)
Nameanotherclassoffunctionsforwhichtheconclusionof
(a)
isvalid.
(c)
Giveanexamplewheres.P/andS.P/arenotRiemannsumsoff overP.
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Section3.1
DefinitionoftheIntegral
127
9.
Find
R
1
0
f.x/dxand
R
1
0
f.x/dxif
(a)
f.x/D
x
ifxisrational;
x
ifxisirrational:
(b)
f.x/D
1 ifxisrational;
x
ifxisirrational:
10.
Giventhat
R
b
a
exdxexists,evaluateitbyusingtheformula
1CrCr
2
CCr
n
D
1r
nC1
1r
.r¤1/
tocalculatecertainRiemannsums.H
INT
:SeeExercise3.1.3:
11.
Giventhat
R
b
0
sinxdxexists,evaluateitbyusingtheidentity
cos.j1/cos.j C1/D2sinsinj
tocalculatecertainRiemannsums.H
INT
:SeeExercise3.1.3:
12.
Giventhat
R
b
0
cosxdxexists,evaluateitbyusingtheidentity
sin.jC1/sin.j1/D2sincosj
tocalculatecertainRiemannsums.H
INT
:SeeExercise3.1.3:
13.
Showthatifg.x/DxCc(c=constant),then
R
b
a
f.x/dg.x/existsifandonlyif
R
b
a
f.x/dxexists,inwhichcase
Z
b
a
f.x/dg.x/D
Z
b
a
f.x/dx:
14.
Supposethat1<a<d <c<1and
g.x/D
g
1
; a<x<d;
g
2
; d d <x<b;
(g
1
;g
2
Dconstants),
andletg.a/, g.b/, andg.d/ bearbitrary. Supposethatf is s definedonŒa;b,
continuousfromtherightataandfromtheleftatb,andcontinuousatd.Showthat
R
b
a
f.x/dg.x/exists,andfinditsvalue.
15.
Supposethat1 < < a D a
0
< a
1
<   < < a
p
D b < < 1, letg.x/ D g
m
(constant)on.a
m1
;a
m
/,1mp,andletg.a
0
/,g.a
1
/,...,g.a
p
/bearbitrary.
Supposethatf isdefinedonŒa;b,continuousfromtherightataandfromthe
leftatb,andcontinuousata
1
,a
2
,...,a
p1
. Evaluate
R
b
a
f.x/dg.x/. H
INT
:See
Exercise3.1.14:
16. (a)
Giveanexamplewhere
R
b
a
f.x/dg.x/existseventhoughf isunbounded
onŒa;b.(Thus,theanalogofTheorem3.1.2doesnotholdfortheRiemann–
Stieltjesintegral.)
(b)
StateandproveananalogofTheorem3.1.2forthecasewheregisincreasing.
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128 Chapter3
IntegralCalculusofFunctionsofOneVariable
17.
ForthecasewheregisnondecreasingandfisboundedonŒa;b,defineupperand
lowerRiemann–StieltjesintegralsinawayanalogoustoDefinition3.1.3.
3.2EXISTENCEOFTHEINTEGRAL
Thefollowinglemmaisthestartingpointforourstudyoftheintegrabilityofabounded
functionfonaclosedintervalŒa;b.
Lemma3.2.1
Supposethat
jf.x/jM; axb;
(3.2.1)
andletP
0
beapartitionofŒa;bobtainedbyaddingrpointstoapartitionP Dfx
0
;x
1
;:::;x
n
g
ofŒa;b:Then
S.P/S.P
0
/S.P/2MrkPk
(3.2.2)
and
s.P/s.P
0
/s.P/C2MrkPk:
(3.2.3)
Proof
Wewillprove(3.2.2)and leavetheproofof(3.2.3)toyou(Exercise 3.2.1).
Firstsupposethatr D1,soP
0
isobtainedbyaddingonepointctothepartitionP D
fx
0
;x
1
;:::;x
n
g;thenx
i1
< c < < x
i
forsomei inf1;2;:::;ng. Ifj ¤ ¤ i,theprod-
uctM
j
.x
j
x
j1
/appearsinbothS.P/andS.P
0
/andcancels outofthedifference
S.P/S.P
0
/.Therefore,if
M
i1
D
sup
x
i1
xc
f.x/ and M
i2
D
sup
cxx
i
f.x/;
then
S.P/S.P
0
/DM
i
.x
i
x
i1
/M
i1
.cx
i1
/M
i2
.x
i
c/
D.M
i
M
i1
/.cx
i1
/C.M
i
M
i2
/.x
i
c/:
(3.2.4)
Since(3.2.1)impliesthat
0M
i
M
ir
2M; rD1;2;
(3.2.4)impliesthat
0S.P/S.P
0
/2M.x
i
x
i1
/2MkPk:
Thisproves(3.2.2)forrD1.
Nowsupposethatr>1andP
0
isobtainedbyaddingpointsc
1
,c
2
,...,c
r
toP. Let
P
.0/
D P P and,forj   1,letP
.j/
bethepartitionofŒa;bobtainedbyaddingc
j
to
P
.j1/
.Thentheresultjustprovedimpliesthat
0S.P
.j1/
/S.P
.j/
/2MkP
.j1/
k; 1j j r:
Section3.2
ExistenceoftheIntegral
129
Addingtheseinequalitiesandtakingaccountofcancellationsyields
0S.P
.0/
/S.P
.r/
/2M.kP
.0/
kCkP
.1/
kCCkP
.r1/
k/:
(3.2.5)
SinceP
.0/
DP,P
.r/
DP
0
,andkP
.k/
kkP
.k1/
kfor1kr1,(3.2.5)implies
that
0S.P/S.P
0
/2MrkPk;
whichisequivalentto(3.2.2).
Theorem3.2.2
Iff isboundedonŒa;b;then
Z
b
a
f.x/dx
Z
b
a
f.x/dx:
(3.2.6)
Proof
SupposethatP
1
andP
2
arepartitionsofŒa;bandP
0
isarefinementofboth.
LettingP DP
1
in(3.2.3)andP DP
2
in(3.2.2)showsthat
s.P
1
/s.P
0
/ and S.P
0
/S.P
2
/:
Sinces.P
0
/S.P
0
/,thisimpliesthats.P
1
/S.P
2
/. Thus,everylowersumisalower
boundforthesetofalluppersums.Since
R
b
a
f.x/dxistheinfimumofthisset,itfollows
that
s.P
1
/
Z
b
a
f.x/dx
foreverypartitionP
1
ofŒa;b.Thismeansthat
R
b
a
f.x/dxisanupperboundfortheset
ofalllowersums.Since
R
b
a
f.x/dxisthesupremumofthisset,thisimplies(3.2.6).
Theorem3.2.3
Iff isintegrableonŒa;b;then
Z
b
a
f.x/dxD
Z
b
a
f.x/dxD
Z
b
a
f.x/dx:
Proof
Weprovethat
R
b
a
f.x/dxD
R
b
a
f.x/dxandleaveittoyoutoshowthat
R
b
a
f.x/dxD
R
b
a
f.x/dx(Exercise3.2.2).
SupposethatPisapartitionofŒa;bandisaRiemannsumoff overP.Since
Z
b
a
f.x/dx
Z
b
a
f.x/dxD
Z
b
a
f.x/dxS.P/
!
C.S.P//
C

Z
b
a
f.x/dx
!
;
130 Chapter3
IntegralCalculusofFunctionsofOneVariable
thetriangleinequalityimpliesthat
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
f.x/dx
Z
b
a
f.x/dx
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/j
C
ˇ
ˇ
ˇ
ˇ
ˇ

Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
:
(3.2.7)
Nowsupposethat>0.FromDefinition3.1.3,thereisapartitionP
0
ofŒa;bsuchthat
Z
b
a
f.x/dxS.P
0
/<
Z
b
a
f.x/dxC
3
:
(3.2.8)
FromDefinition3.1.1,thereisaı>0suchthat
ˇ
ˇ
ˇ
ˇ
ˇ

Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
<
3
(3.2.9)
ifkPk<ı.NowsupposethatkPk<ıandPisarefinementofP
0
.SinceS.P/S.P
0
/
byLemma3.2.1,(3.2.8)impliesthat
Z
b
a
f.x/dxS.P/<
Z
b
a
f.x/dxC
3
;
so
ˇ
ˇ
ˇ
ˇ
ˇ
S.P/
Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
<
3
(3.2.10)
inadditionto(3.2.9).Now(3.2.7),(3.2.9),and(3.2.10)implythat
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
f.x/dx
Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
<
2
3
CjS.P/j
(3.2.11)
foreveryRiemannsumoff overP. SinceS.P/isthesupremumoftheseRiemann
sums(Theorem3.1.4),wemaychoosesothat
jS.P/j<
3
:
Now(3.2.11)impliesthat
ˇ
ˇ
ˇ
ˇ
ˇ
Z
b
a
f.x/dx
Z
b
a
f.x/dx
ˇ
ˇ
ˇ
ˇ
ˇ
<:
Sinceisanarbitrarypositivenumber,itfollowsthat
Z
b
a
f.x/dxD
Z
b
a
f.x/dx:
Section3.2
ExistenceoftheIntegral
131
Lemma3.2.4
Iff isboundedonŒa;band>0;thereisaı>0suchthat
Z
b
a
f.x/dxS.P/<
Z
b
a
f.x/dxC
(3.2.12)
and
Z
b
a
f.x/dxs.P/>
Z
b
a
f.x/dx
ifkPk<ı.
Proof
Weshowthat(3.2.12)holdsifkPkissufficientlysmall,andleavetherestofthe
prooftoyou(Exercise3.2.3).
Thefirstinequalityin(3.2.12)followsimmediatelyfromDefinition3.1.3. Toestablish
thesecondinequality,supposethatjf.x/jKifaxb.FromDefinition3.1.3,there
isapartitionP
0
Dfx
0
;x
1
;:::;x
rC1
gofŒa;bsuchthat
S.P
0
/<
Z
b
a
f.x/dxC
2
:
(3.2.13)
IfP isanypartitionofŒa;b,letP
0
beconstructedfromthepartitionpointsofP
0
andP.
Then
S.P
0
/S.P
0
/;
(3.2.14)
byLemma 3.2.1. SinceP
0
isobtainedbyaddingatmostr pointstoP,Lemma3.2.1
impliesthat
S.P
0
/S.P/2KrkPk:
(3.2.15)
Now(3.2.13),(3.2.14),and(3.2.15)implythat
S.P/S.P
0
/C2KrkPk
S.P
0
/C2KrkPk
<
Z
b
a
f.x/dxC
2
C2KrkPk:
Therefore,(3.2.12)holdsif
kPk<ıD
4Kr
:
132 Chapter3
IntegralCalculusofFunctionsofOneVariable
Theorem3.2.5
Iff isboundedonŒa;band
Z
b
a
f.x/dxD
Z
b
a
f.x/dxDL;
(3.2.16)
thenf isintegrableonŒa;band
Z
b
a
f.x/dxDL:
(3.2.17)
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