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/:
FdoesnothaveaderivativeatxD1,wheref isdiscontinuous,since
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
/:
Pdf email link - insert, remove PDF links in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Free C# example code is offered for users to edit PDF document hyperlink (url), like inserting and deleting
add links to pdf acrobat; add links in pdf
Pdf email link - VB.NET PDF url edit library: insert, remove PDF links in vb.net, ASP.NET, MVC, Ajax, WinForms, WPF
Help to Insert a Hyperlink to Specified PDF Document Page
add a link to a pdf in acrobat; pdf email link
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:
RasterEdge.com General FAQs for Products
copy and email the secure download link to the assistance, please contact us via email (support@rasteredge & profession imaging controls, PDF document, image to
pdf link to attached file; adding hyperlinks to pdf
RasterEdge Product Licensing Discount
s). After confirming the informations provided, we will send you an email that contains price(s) at a discount and the online order link for new licensing.
add links to pdf; pdf links
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)
RasterEdge Product Renewal and Update
4. Order email. Our support team will send you the purchase link. HTML5 Viewer for .NET; XDoc.Windows Viewer for .NET; XDoc.Converter for .NET; XDoc.PDF for .NET;
adding hyperlinks to pdf files; add hyperlink pdf
VB.NET Create PDF from PowerPoint Library to convert pptx, ppt to
Link: Edit URL. Bookmark: Edit Bookmark. Metadata: Edit, Delete Metadata. Form Process. Create PDF file from PowerPoint free online without email.
adding links to pdf in preview; adding hyperlinks to pdf documents
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//
VB.NET Create PDF from Word Library to convert docx, doc to PDF in
Link: Edit URL. Bookmark: Edit Bookmark. Metadata: Edit, Delete Metadata. Form Process. Free online Word to PDF converter without email.
add hyperlink to pdf; convert a word document to pdf with hyperlinks
VB.NET Create PDF from Excel Library to convert xlsx, xls to PDF
Link: Edit URL. Bookmark: Edit Bookmark. Metadata: Edit, Delete Metadata. Form Process. Convert Excel to PDF document free online without email.
adding links to pdf; add hyperlink pdf file
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
:
VB.NET PDF Convert to Word SDK: Convert PDF to Word library in vb.
Create editable Word file online without email. Supports transfer from password protected PDF. VB.NET class source code for .NET framework.
add hyperlink to pdf in; add hyperlink to pdf online
C# PDF Convert to Word SDK: Convert PDF to Word library in C#.net
and .docx. Create editable Word file online without email. Password protected PDF file can be printed to Word for mail merge. C# source
add a link to a pdf; clickable links in pdf
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
;
C# Create PDF from Excel Library to convert xlsx, xls to PDF in C#
Export PDF from Excel with cell border or no border. Free online Excel to PDF converter without email. Quick integrate online C# source code into .NET class.
adding a link to a pdf in preview; add link to pdf file
C# Create PDF from PowerPoint Library to convert pptx, ppt to PDF
application. Free online PowerPoint to PDF converter without email. C# source code is provided for .NET WinForms class. Evaluation
add a link to a pdf file; pdf link to specific page
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.
SinceisintegrableonŒc;d,thiscanbemadeassmallaswepleasebychoosingk
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.
Pj<=3inadditionto(3.3.31)yields
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.
Prove:Iff isintegrableonŒa;bandaDa
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.
152 Chapter3
IntegralCalculusofFunctionsofOneVariable
16.
Thesymbol
R
f.x/dxdenotesanantiderivativeoff. AplausibleanalogofThe-
orem3.3.1wouldstatethatiff andghaveantiderivativesonŒa;b,thensodoes
f Cg,whichistrue,and
Z
.f Cg/.x/dxD
Z
f.x/dxC
Z
g.x/dx:
.A/
However,thisisnottrueintheusualsense.
(a)
Whynot?
(b)
Stateacorrectinterpretationof(A).
17.
(SeeExercise3.3.16.)Formulateavalidinterpretationoftherelation
Z
.cf/.x/dxDc
Z
f.x/dx .c¤0/:
IsyourinterpretationvalidifcD0?
18. (a)
Letf
.nC1/
beintegrableonŒa;b.Showthat
f.b/D
Xn
rD0
f
.r/
.a/
.ba/
r
C
1
Z
b
a
f
.nC1/
.t/.bt/
n
dt:
H
INT
:Integratebypartsanduseinduction:
(b)
Whatistheconnectionbetween
(a)
andTheorem2.5.5?
19.
InadditiontotheassumptionsofTheorem3.3.16,supposethatf.a/D0,f 60,
andg.x/>0.a<x <b/. ShowthatthereisonlyonepointcinŒa;bwiththe
propertystatedinTheorem3.3.16.H
INT
:UseExercise3.3.13:
20.
AssumingthatTheorem 3.3.18istrueundertheadditionalassumptionthatf is
nonnegativeonŒa;b,showthatitistruewithoutthisassumption.
21.
AssumingthattheconclusionofTheorem3.3.18istrueifisnondecreasing,show
thatitistrueifisnonincreasing.H
INT
:UseExercise3.1.6:
22.
Supposeg
0
isintegrableandf iscontinuousonŒa;b. . Showthat
R
b
a
f.x/dg.x/
existsandequals
R
b
a
f.x/g
0
.x/dx.
23.
Supposefandg
00
areboundedandfg
0
isintegrableonŒa;b.Showthat
R
b
a
f.x/dg.x/
existsandequals
R
b
a
f.x/g
0
.x/dx.H
INT
:UseTheorem2.5.4:
3.4IMPROPERINTEGRALS
Sofarwehaveconfinedourstudyoftheintegraltoboundedfunctionsonfiniteclosed
intervals.Thiswasforgoodreasons:
 FromTheorem3.1.2,anunboundedfunctioncannotbeintegrableonafiniteclosed
interval.
Documents you may be interested
Documents you may be interested