c# pdf free : C# read pdf from url SDK Library API .net wpf azure sharepoint TRENCH_REAL_ANALYSIS12-part229

CHAPTER3
IntegralCalculusof
FunctionsofOne Variable
INTHISCHAPTERwediscusstheRiemannonafiniteintervalŒa;b,andimproperinte-
gralsinwhicheitherthefunctionortheintervalofintegrationisunbounded.
SECTION3.1beginswiththedefinitionoftheRiemannintegralandpresentsthegeo-
metricalinterpretationoftheRiemannintegralastheareaunderacurve. Weshowthat
anunboundedfunctioncannotbeRiemannintegrable. Thenwedefineupperandlower
sumsandupperandlowerintegralsofaboundedfunction.Thesectionconcludeswiththe
definitionoftheRiemann–Stieltjesintegral.
SECTION3.2presentsnecessaryandsufficientconditionsfortheexistenceoftheRiemann
integralintermsofupperandlowersumsandupperandlowerintegrals. Weshowthat
continuousfunctionsandboundedmonotonicfunctionsareRiemannintegrable.
SECTION3.3beginswithproofsthatthesumandproductofRiemannintegrablefunctions
areintegrable,andthatjfjisRiemannintegrableiff isRiemannintegrable.Othertopics
coveredincludethefirstmeanvaluetheoremforintegrals,antiderivatives,thefundamental
theoremofcalculus,changeofvariables,integrationbyparts,andthesecondmeanvalue
theoremforintegrals.
SECTION3.4presentsacomprehensivediscussionofimproperintegrals. Conceptsde-
finedandconsideredincludeabsoluteandconditionalconvergenceofanimproperintegral,
Dirichlet’stest,andchangeofvariableinanimproperintegral.
SECTION3.5definesthenotionofasetwithLebesguemeasure zero, andpresentsa
necessaryandsufficientconditionforaboundedfunctionf tobeRiemannintegrableon
anintervalŒa;b;namely,thatthediscontinuitiesoff formasetwithLebesguemasure
zero.
3.1DEFINITIONOF THEINTEGRAL
TheintegralthatyoustudiedincalculusistheRiemannintegral,namedaftertheGerman
mathematicianBernhardRiemann,whoprovidedarigorousformulationoftheintegralto
113
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114 Chapter3
IntegralCalculusofFunctionsofOneVariable
replacetheintuitivenotionduetoNewtonandLeibniz.SinceRiemann’stime,otherkinds
ofintegralshavebeendefinedandstudied;however, theyareallgeneralizationsofthe
Riemannintegral,anditishardlypossibletounderstandthemorappreciatethereasonsfor
developingthemwithoutathoroughunderstandingoftheRiemannintegral.Inthissection
wedealwithfunctionsdefinedonafiniteintervalŒa;b. ApartitionofŒa;bisasetof
subintervals
Œx
0
;x
1
;Œx
1
;x
2
;:::;Œx
n1
;x
n
;
(3.1.1)
where
aDx
0
<x
1
<x
n
Db:
(3.1.2)
Thus,anysetofnC1pointssatisfying(3.1.2)definesapartitionP ofŒa;b,whichwe
denoteby
P Dfx
0
;x
1
;:::;x
n
g:
Thepointsx
0
,x
1
,...,x
n
arethepartitionpointsofP. Thelargestofthelengthsofthe
subintervals(3.1.1)isthenormofP,writtenaskPk;thus,
kPkD max
1in
.x
i
x
i1
/:
IfP andP
0
arepartitionsofŒa;b,thenP
0
isarefinementofP ifeverypartitionpoint
ofPisalsoapartitionpointofP
0
;thatis,ifP
0
isobtainedbyinsertingadditionalpoints
betweenthoseofP.Iff isdefinedonŒa;b,thenasum
 D
Xn
jD1
f.c
j
/.x
j
x
j1
/;
where
x
j1
c
j
x
j
; 1j j n;
isaRiemannsumoff overthepartitionP P Dfx
0
;x
1
;:::;x
n
g.(Occasionallywewillsay
moresimplythatisaRiemannsumoff overŒa;b.)Sincec
j
canbechosenarbitrarily
inŒx
j
;x
j1
,thereareinfinitelymanyRiemannsumsforagivenfunctionf overagiven
partitionP.
Definition3.1.1
Letf bedefinedonŒa;b.Wesaythatf isRiemannintegrableon
Œa;bifthereisanumberLwiththefollowingproperty:Forevery>0,thereisaı>0
suchthat
jLj<
ifisanyRiemannsumoff overapartitionP P ofŒa;bsuchthatkPk<ı.Inthiscase,
wesaythatListheRiemannintegraloffoverŒa;b,andwrite
Z
b
a
f.x/dxDL:
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Section3.1
DefinitionoftheIntegral
115
Weleaveittoyou(Exercise3.1.1)toshowthat
R
b
a
f.x/dxisunique,ifitexists;thatis,
therecannotbemorethanonenumberLthatsatisfiesDefinition3.1.1.
Forbrevitywewillsay“integrable”and“integral”whenwemean“Riemannintegrable”
and“Riemannintegral.” Sayingthat
R
b
a
f.x/dxexistsisequivalenttosayingthatf is
integrableonŒa;b.
Example3.1.1
If
f.x/D1; axb;
then
Xn
jD1
f.c
j
/.x
j
x
j1
/D
Xn
jD1
.x
j
x
j1
/:
Mostofthetermsinthesumontherightcancelinpairs;thatis,
n
X
jD1
.x
j
x
j1
/D.x
1
x
0
/C.x
2
x
1
/CC.x
n
x
n1
/
Dx
0
C.x
1
x
1
/C.x
2
x
2
/CC.x
n1
x
n1
/Cx
n
Dx
n
x
0
Dba:
Thus,everyRiemannsumoff overanypartitionofŒa;bequalsba,so
Z
b
a
dxDba:
Example3.1.2
Riemannsumsforthefunction
f.x/Dx; axb;
areoftheform
D
Xn
jD1
c
j
.x
j
x
j1
/:
(3.1.3)
Sincex
j1
c
j
x
j
and.x
j
Cx
j1
/=2isthemidpointofŒx
j1
;x
j
,wecanwrite
c
j
D
x
j
Cx
j1
2
Cd
j
;
(3.1.4)
where
jd
j
j
x
j
x
j1
2
kPk
2
:
(3.1.5)
Substituting(3.1.4)into(3.1.3)yields
D
n
X
jD1
x
j
Cx
j1
2
.x
j
x
j1
/C
n
X
jD1
d
j
.x
j
x
j1
/
D
1
2
Xn
jD1
.x
2
j
x
2
j1
/C
Xn
jD1
d
j
.x
j
x
j1
/:
(3.1.6)
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116 Chapter3
IntegralCalculusofFunctionsofOneVariable
BecauseofcancellationslikethoseinExample3.1.1,
Xn
jD1
.x
2
j
x
2
j1
/Db
2
a
2
;
so(3.1.6)canberewrittenas
D
b
2
a
2
2
C
Xn
jD1
d
j
.x
j
x
j1
/:
Hence,
ˇ
ˇ
ˇ
ˇ

b2a2
2
ˇ
ˇ
ˇ
ˇ
Xn
jD1
jd
j
j.x
j
x
j1
/
kPk
2
Xn
jD1
.x
j
x
j1
/ (see(3.1.5))
D
kPk
2
.ba/:
Therefore,everyRiemannsumoff overapartitionP P ofŒa;bsatisfies
ˇ
ˇ
ˇ
ˇ

b
2
a
2
2
ˇ
ˇ
ˇ
ˇ
< if kPk<ıD
2
ba
:
Hence,
Z
b
a
xdxD
b2a2
2
:
TheIntegral astheAreaUnderaCurve
Animportantapplicationoftheintegral,indeed,theoneinvariablyusedtomotivateits
definition,isthecomputationoftheareaboundedbyacurveyDf.x/,thex-axis,and
thelinesxDaandxDb(“theareaunderthecurve”),asinFigure3.1.1.
y
x
b
a
y = f(x)
Figure3.1.1
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Section3.1
DefinitionoftheIntegral
117
Forsimplicity,supposethatf.x/>0.Thenf.c
j
/.x
j
x
j1
/istheareaofarectangle
withbasex
j
x
j1
andheightf.c
j
/,sotheRiemannsum
n
X
jD1
f.c
j
/.x
j
x
j1
/
canbeinterpretedasthesumoftheareasofrectanglesrelatedtothecurveyDf.x/,as
showninFigure3.1.2.
y
x
a c
1
x
1
x
2
c
2
x
3
c
3
c
4
b
y = f(x)
Figure3.1.2
Anapparentlyplausibleargument,thattheRiemannsumsapproximatetheareaunder
thecurvemoreandmorecloselyasthenumberofrectanglesincreasesandthelargestof
theirwidthsismadesmaller,seemstosupporttheassertionthat
R
b
a
f.x/dx equalsthe
areaunderthecurve. ThisargumentisusefulasamotivationforDefinition3.1.1,which
withoutitwouldseemmysterious. Nevertheless,thelogicisincorrect,sinceitisbased
ontheassumptionthattheareaunderthecurvehasbeenpreviouslydefinedinsomeother
way.Althoughthisistrueforcertaincurvessuchas,forexample,thoseconsistingofline
segmentsorcirculararcs,itisnottrueingeneral.Infact,theareaunderamorecomplicated
curveisdefinedtobeequaltotheintegral,iftheintegralexists.Thatthisnewdefinitionis
consistentwiththeoldone,wherethelatterapplies,isevidencethattheintegralprovides
ausefulgeneralizationofthedefinitionofarea.
Example3.1.3
Letf.x/Dx,1x2(Figure3.1.3,page118).Theregionunder
thecurveconsistsofasquareofunitarea,surmountedbyatriangleofarea1=2;thus,the
areaoftheregionis3=2.FromExample3.1.2,
Z
2
1
xdxD
1
2
.2
2
1
2
/D
3
2
;
sotheintegralequalstheareaunderthecurve.
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118 Chapter3
IntegralCalculusofFunctionsofOneVariable
y
x
2
1
y = x
Figure3.1.3
y
x
y = x2
2
1
Figure3.1.4
Example3.1.4
If
f.x/Dx
2
; 1x2
(Figure3.1.4),then
Z
2
1
f.x/dxD
1
3
.2
3
1
3
/D
7
3
(Exercise3.1.4),sowesaythattheareaunderthecurveis7=3.However,thisisthedefini-
tionofthearearatherthanaconfirmationofapreviouslyknownfact,asinExample3.1.3.
Section3.1
DefinitionoftheIntegral
119
Theorem3.1.2
Iff isunboundedonŒa;b;thenf isnotintegrableonŒa;b:
Proof
Wewillshowthatiff isunboundedonŒa;b,P isanypartitionofŒa;b,and
M>0,thenthereareRiemannsumsand
0
offoverPsuchthat
j
0
jM:
(3.1.7)
Weleaveittoyou(Exercise3.1.2)tocompletetheproofbyshowingfromthisthatf
cannotsatisfyDefinition3.1.1.
Let
D
Xn
jD1
f.c
j
/.x
j
x
j1
/
bea Riemann sum off overa a partitionP P ofŒa;b. Theremust t beanintegeri i in
f1;2;:::;ngsuchthat
jf.c/f.c
i
/j
M
x
i
x
i1
(3.1.8)
forsomecinŒx
i1
x
i
,becauseiftherewerenotso,wewouldhave
jf.x/f.c
j
/j<
M
x
j
x
j1
; x
j1
xx
j
; 1j j n:
Then
jf.x/jDjf.c
j
/Cf.x/f.c
j
/jjf.c
j
/jCjf.x/f.c
j
/j
jf.c
j
/jC
M
x
j
x
j1
; x
j1
xx
j
; 1j j n:
whichimpliesthat
jf.x/j max
1jn
jf.c
j
/jC
M
x
j
x
j1
; axb;
contradictingtheassumptionthatf isunboundedonŒa;b.
Nowsupposethatcsatisfies(3.1.8),andconsidertheRiemannsum
0
D
n
X
jD1
f.c
0
j
/.x
j
x
j1
/
overthesamepartitionP,where
c
0
j
D
c
j
; j j ¤i;
c; j j Di:
120 Chapter3
IntegralCalculusofFunctionsofOneVariable
Since
j
0
jDjf.c/f.c
i
/j.x
i
x
i1
/;
(3.1.8)implies(3.1.7).
UpperandLowerIntegrals
BecauseofTheorem3.1.2,weconsideronlyboundedfunctionsthroughouttherestofthis
section.
ToprovedirectlyfromDefinition3.1.1that
R
b
a
f.x/dxexists,itisnecessarytodiscover
itsvalueLinonewayoranotherandtoshowthatLhasthepropertiesrequiredbythe
definition.Foraspecificfunctionitmayhappenthatthiscanbedonebystraightforward
calculation,asinExamples3.1.1and3.1.2.However,thisisnotsoiftheobjectiveistofind
generalconditionswhichimplythat
R
b
a
f.x/dxexists.Thefollowingapproachavoidsthe
difficultyofhavingtodiscoverLinadvance,withoutknowingwhetheritexistsinthefirst
place,andrequiresonlythatwecomparetwonumbersthatmustexistiff isboundedon
Œa;b.Wewillseethat
R
b
a
f.x/dxexistsifandonlyifthesetwonumbersareequal.
Definition3.1.3
Iff isboundedonŒa;bandP Dfx
0
;x
1
;:::;x
n
gisapartitionof
Œa;b,let
M
j
D
sup
x
j1
xx
j
f.x/
and
m
j
D
inf
x
j1
xx
j
f.x/:
TheuppersumoffoverP is
S.P/D
Xn
jD1
M
j
.x
j
x
j1
/;
andtheupperintegraloffover,Œa;b,denotedby
Z
b
a
f.x/dx;
istheinfimumofalluppersums.Thelowersumoff overP P is
s.P/D
Xn
jD1
m
j
.x
j
x
j1
/;
andthelowerintegraloff overŒa;b,denotedby
Z
b
a
f.x/dx;
isthesupremumofalllowersums.
Section3.1
DefinitionoftheIntegral
121
Ifmf.x/MforallxinŒa;b,then
m.ba/s.P/S.P/M.ba/
foreverypartitionP;thus,thesetofuppersumsoff overallpartitionsP P ofŒa;bis
bounded, as isthesetoflowersums. Therefore, , Theorems1.1.3and1.1.8implythat
R
b
a
f.x/dxand
R
b
a
f.x/dxexist,areunique,andsatisfytheinequalities
m.ba/
Z
b
a
f.x/dxM.ba/
and
m.ba/
Z
b
a
f.x/dxM.ba/:
Theorem3.1.4
Letf beboundedonŒa;b,andletPbeapartitionofŒa;b:Then
(a)
TheuppersumS.P/off overP P isthesupremumofthesetofallRiemannsumsof
f overP:
(b)
Thelowersums.P/off overPistheinfimumofthesetofallRiemannsumsoff
overP:
Proof (a)
IfP Dfx
0
;x
1
;:::;x
n
g,then
S.P/D
Xn
jD1
M
j
.x
j
x
j1
/;
where
M
j
D
sup
x
j1
xx
j
f.x/:
AnarbitraryRiemannsumoff overP P isoftheform
 D
Xn
jD1
f.c
j
/.x
j
x
j1
/;
wherex
j1
c
j
x
j
.Sincef.c
j
/M
j
,itfollowsthatS.P/.
Nowlet>0andchoose
c
j
inŒx
j1
;x
j
sothat
f.
c
j
/>M
j
n.x
j
x
j1
/
; 1j j n:
TheRiemannsumproducedinthiswayis
D
Xn
jD1
f.
c
j
/.x
j
x
j1
/>
Xn
jD1
M
j
n.x
j
x
j1
/
/
.x
j
x
j1
/DS.P/:
NowTheorem1.1.3impliesthatS.P/isthesupremumofthesetofRiemannsumsoff
overP.
(b)
Exercise3.1.7.
122 Chapter3
IntegralCalculusofFunctionsofOneVariable
Example3.1.5
Let
f.x/D
0
ifxisirrational;
1
ifxisrational;
andP Dfx
0
;x
1
;:::;x
n
gbeapartitionofŒa;b.Sinceeveryintervalcontainsbothratio-
nalandirrationalnumbers(Theorems1.1.6and1.1.7),
m
j
D0 and M
j
D1; 1j j n:
Hence,
S.P/D
Xn
jD1
1.x
j
x
j1
/Dba
and
s.P/D
Xn
jD1
0.x
j
x
j1
/D0:
Sincealluppersumsequalbaandalllowersumsequal0,Definition3.1.3impliesthat
Z
b
a
f.x/dxDba and
Z
b
a
f.x/dxD0:
Example3.1.6
LetfbedefinedonŒ1;2byf.x/D0ifxisirrationalandf.p=q/D
1=qifp andqarepositiveintegerswithnocommonfactors(Exercise2.2.7). IfP D
fx
0
;x
1
;:::;x
n
gisanypartitionofŒ1;2,thenm
j
D0,1j n,sos.P/D0;hence,
Z
2
1
f.x/dxD0:
Wenowshowthat
Z
2
1
f.x/dxD0
(3.1.9)
also.SinceS.P/>0foreveryP,Definition3.1.3impliesthat
Z
2
1
f.x/dx0;
soweneedonlyshowthat
Z
2
1
f.x/dx0;
whichwillfollowifweshowthatnopositivenumberislessthaneveryuppersum.Tothis
end,weobservethatif0<<2,thenf.x/=2foronlyfinitelymanyvaluesofxin
Œ1;2.
LetkbethenumberofsuchpointsandletP
0
beapartitionofŒ1;2suchthat
kP
0
k<
2k
:
(3.1.10)
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