微積分第五章課件

Chapter5Integrals

Chapter5Integrals

15.1AreasandDistance5.2TheDefiniteIntegral5.3TheFundamentalTheoremofCalculus5.4IndefiniteIntegralsandtheNetChangeTheorem5.5TheSubstitutionRule5.6TheLogarithmDefinedasanIntegral5.1AreasandDistance5.2T25.1AreasandDistances5.1AreasandDistances3Theareaproblem1.AreasofcurvedtrapezoidSupposethecurvedtrapezoidisboundedbyandFindtheareaofA.AreaofrectangleAreaoftrapezoidTheareaproblem1.Areasof4Method:1)

Partition:UsethelinestodivideAintosmallcurvedtrapezoid;2)

Approximation:Base:Height:WecanapproximateSmallrectangle:Method:1)Partition:Usetheli53)Sum:4)Limit:Letthentheareaofthecurvedtrapezoidis:Riemannsum3)Sum:4)Limit:Letthenthear62.DistanceandFindthetotaldistances.Method:1)Partition:2)Approximation:WegetSupposeSupposethedistanceover2.DistanceandFindthetotald73)Sum:4)Limit:Commoncharacteristicoftheaboveproblemsis:1)Theprocessisthesame:“Partition,Approximation,Sum,Limit”2)Thelimitformsarethesame:BotharethelimitsofRiemannsumsRiemannsum3)Sum:4)Limit:Commoncharact85.2Thedefinitionofintegral5.2Thedefinitionofintegra9ThedefinitionofintegralalwaystendstoI,WesaythatIistheintegralof

ThatisThenwesaythatf(x)isintegrableon[a,b].denoteitbyThedefinitionofintegralalw10where:integralsignf(x):integranda:lowerlimitofintegrationb:upperlimitofintegrationTheprocedureofcalculatinganintegraliscalledintegration.where:integralsignf(x):int11Caution:Thedefiniteintegralisanumber;itdoesnotdependonx.Infact,wecoulduseanyletterinplaceofxwithoutchangingthevalueoftheintegral:wherexisadummyvariable.Caution:Thedefiniteintegral12Geometricinterpretation::theareaoftheregion:thenegativeoftheareaoftheregion:thealgebraicsumoftheorientedarea,orthenetarea.Geometricinterpretation::the13Theorem1.Theorem2.andhaveonlyfinitediscontinuitieson[a,b]Thesufficientconditionofintegrability:isintegrableon

isintegrableon

Theorem1.Theorem2.andhaveonl14Example1.UsethedefinitiontoevaluateSolution:UseWechoosetodivide[0,1]intonsubintervalsofequalwidth.Example1.Usethedefinition15微積分第五章課件16Example2.UsethedefinitiontoevaluateSolution:UseWechoosetodivide[1,3]intonsubintervalsofequalwidth.Example2.Usethedefinition17微積分第五章課件18Solution:Usetheintegraltodenotethefollowinglimits:Solution:Usetheintegraltod19Usethegeometricmeaningtoevaluate:Usethegeometricmeaningtoe20TheMidpointRuleTheMidpointRule21PropertiesofthedefiniteintegralWhenwedefinedthedefiniteintegral.weimplicitlyassumedthata<b.ButthedefinitionasalimitofRiemannsumsmakessenseevenifa>b.Noticethatifwereverseaandb,thenchange(b-a)/nfromto(a-b)/n.ThereforePropertiesofthedefiniteint22微積分第五章課件23Propertiesoftheintegral(kisaconstant)Proof:(cisaconstant)Propertiesoftheintegral(k24

Example:Usethepropertiesofintegralstoevaluate

Example:Usethepropert25微積分第五章課件26ComparisonPropertiesoftheintegral:ComparisonPropertiesofthei27ThenexttheoremiscalledMeanValueTheoremfor

DefiniteIntegrals.Itsgeometricinterpretationisthat,fora

continuouspositivef(x)on[a,b],thereisanumbercin[a,b]suchthattherectanglewithbase[a,b]andheightf(c)hasthesameareaastheregionunderthegraphoff(x)fromatob.oabxycf(x)TheoremIff(x)iscontinuouson[a,b],thenthereexistsatleastanumbercin(a,b)suchthatThenexttheoremiscalledMea28Thenumberiscalledtheaveragevalueoff(x)on[a,b].Proof

Iff(x)isaconstantfunction,theresultistrue.Nextweassumethatf(x)isnotaconstantfunction.Sincef(x)iscontinuouson[a,b],f(x)takesontheminimumandthemaximumvalueson[a,b].Letf(u)=mandf(v)=Mbetheminimumandthemaximumvaluesoff(x)on[a,b],respectively.Thenm<f(x)<Mforsomexin[a,b]becausef(x)isnotconstant.Therefore,wehaveItfollowsthat

Thenumber29Theprecedinginequalitiesindicatethatthenumber

isbetweenm=f(u)andM=f(v).ThustheIntermediateValueTheoremleadsthatthereisanumbercbetweenuandvsuchthatMultiplyingthebothsidesbyb-agivestheconclusionofthetheorem.Theprecedinginequalitiesin305.3TheFundamentalTheorem

ofCalculus5.3TheFundamentalTheorem31TheFundamentalTheoremofCalculusThefundamentalTheoremofCalculusisappropriatelynamedbecauseitestablishesaconnectionbetweenthetwobranchestoCalculus:differentialcalculusandintegralcalculus.Differentialcalculusarosefromthetangentproblem,whereasintegralcalculusarosefromaseeminglyunrelatedproblem,theareaproblem.TheFundamentalTheoremofCalculusgivesthepreciserelationshipbetweenthederivativeandtheintegral.TheFundamentalTheoremofCa32Newton’steacheratCambridge,IssacBarrowdiscoveredthatthetwoproblemareactuallycloselyrelated.Infact,herealizedthatdifferentiationandintegrationareinverseprocesses.ItwasNewtonandLeibnizwhoexploitedthispreciserelationshipanduseittodevelopcalculusintoasystematicmathematicalmethod.Inparticular,theysawthattheFundamentalTheoremenabledthemtocomputeareasandintegralsveryeasilywithouthavingtocomputethemaslimitsofsums.Newton’steacheratCambridge,33ThefirstpartoftheFundamentalTheoremdealswithfunctionsdefinedbyanequationoftheformwherefisacontinuousfunctionon[a,b]andxvariesbetweenaandb.Observethatgdependsonlyonx,whichappearsasthevariableupperlimitintheintegral.Ifxisafixednumber,thentheintegralisadefinitenumber.Ifwethenletxvary,thenumberalsovariesanddefinesafunctionofxbyg(x).baxf(t)ThefirstpartoftheFundamen34TheFundamentalTheoremofCalculus,PartIProof:thenTheFundamentalTheoremofCal35微積分第五章課件36微積分第五章課件37微積分第五章課件38TheFundamentalTheoremofCalculus,Part2Proof:Accordingtopart1,sosowehavedenoteTheFundamentalTheoremofCal39微積分第五章課件40微積分第五章課件41Why?Why?42Summary:WeendthissectionbybringingtogetherthetwopartsoftheFundamentalTheorem.Summary:Weendthissectionby435.4IndefiniteIntegrals

andtheNetChangeTheorem5.4IndefiniteIntegrals44IndefiniteIntegrals

andtheNetChangeTheoremAnindefiniteintegraloff(x)representanentirelyfamilyoffunctionswhosederivativeisf(x)

andisdenotedbySupposethatF(x)isanantiderivativeoff(x),thenaccordingtothetheoremin4.2,weknowthatanyantiderivativeG(x)off(x)canbewrittenas

G(x)=F(x)+CIndefiniteIntegrals

and45Then:Caution:1)Youshoulddistinguishcarefullybetweendefiniteandindefiniteintegrals.Adefiniteintegralisanumber,whereasanindefiniteintegralisafunction(afamilyoffunctions).2)Theconnectionisthefundamentaltheorem:Then:Caution:46TableofIndefiniteintegralsFromtheabovedefinition,weknownthat:oror(kisaconstant)TableofIndefiniteintegrals47orororor48微積分第五章課件49Example1:Sulotion:=Example2:Solution:=Example1:Sulotion:=50ThepropertiesofIndefiniteIntegralsCorollary:IfthenThepropertiesofIndefiniteI51Example3:Solution:=Example3:Solution:=52Example4:Solution:=Example5:Solution:=Example4:Solution:=Example553Example:

Solution:=Example:Solution:=54Application:TheNetChangeTheoremTheintegralofarateofchangeisthenetchange:Application:TheNetChangeThe55微積分第五章課件56Example6Aparticlemovesalongalinesothatitsvelocityattimetis1)Findthedisplacementoftheparticleduringthetimeperiod2)Findthedistancetraveledduringthetimeperiod.Solution:Example6Aparticlemovesa575.5TheSubstitutionRule5.5TheSubstitutionRule58TheSubstitutionRuleExample:TheSubstitutionRuleExample:59微積分第五章課件60

TheSubstitutionRule:Suppose()TheSubstitutionRule:Suppose61Exampe1

Exampe162Example2

Example263Example3Example364Example4

Example465Example5

Example566(1).(1).67(2).(2).68(3).(3).69Ex6Ex7Ex6Ex770DefiniteIntegralsTheSubstitutionRuleforDefiniteIntegralsIfiscontinuouson[a,b]andfiscontinuousontherangeofu=g(x),thenProof:SupposeFbeanantiderivativeoff.ThenF(g(x))isanantiderivativeofDefiniteIntegralsTheSubstitu71Ex1.

EvaluateSolution:LetthenTheaboveintegral=andEx1.EvaluateSolution:Letthe72Ex2.

EvaluateSolution:

LetthenTheaboveintegral=andEx2.EvaluateSolution:thenThe73Ex3.Proof:(1)If(2)IfEx3.Proof:(1)If(2)If74Question:Question:755.6TheLogarithmDefinedasanIntegral5.6TheLogarithmDefinedasa76TheLogarithmDefinedasanIntegralOurtreatmentofexponentialandlogarithmicfunctionsuntilnowhasreliedonourintuition,whichisbasedonnumericalandvisualevidence.HereweusetheFundamentalTheoremofCalculustogiveanalternativetreatmentthatprovidesasurerfootingforthesefunctions.TheLogarithmDefinedasanI77Definition

ThenaturallogarithmicfunctionisthefunctiondefinedbyBasedontheabovedefinitionofnaturallogarithmicfunction,wewilldeduceallpropertiesoflogarithmfunctions。Asfarasthedetails,welefttoyouasanexercise.DefinitionThenaturallogar78Chapter5Integrals

Chapter5Integrals

795.1AreasandDistance5.2TheDefiniteIntegral5.3TheFundamentalTheoremofCalculus5.4IndefiniteIntegralsandtheNetChangeTheorem5.5TheSubstitutionRule5.6TheLogarithmDefinedasanIntegral5.1AreasandDistance5.2T805.1AreasandDistances5.1AreasandDistances81Theareaproblem1.AreasofcurvedtrapezoidSupposethecurvedtrapezoidisboundedbyandFindtheareaofA.AreaofrectangleAreaoftrapezoidTheareaproblem1.Areasof82Method:1)

Partition:UsethelinestodivideAintosmallcurvedtrapezoid;2)

Approximation:Base:Height:WecanapproximateSmallrectangle:Method:1)Partition:Usetheli833)Sum:4)Limit:Letthentheareaofthecurvedtrapezoidis:Riemannsum3)Sum:4)Limit:Letthenthear842.DistanceandFindthetotaldistances.Method:1)Partition:2)Approximation:WegetSupposeSupposethedistanceover2.DistanceandFindthetotald853)Sum:4)Limit:Commoncharacteristicoftheaboveproblemsis:1)Theprocessisthesame:“Partition,Approximation,Sum,Limit”2)Thelimitformsarethesame:BotharethelimitsofRiemannsumsRiemannsum3)Sum:4)Limit:Commoncharact865.2Thedefinitionofintegral5.2Thedefinitionofintegra87ThedefinitionofintegralalwaystendstoI,WesaythatIistheintegralof

ThatisThenwesaythatf(x)isintegrableon[a,b].denoteitbyThedefinitionofintegralalw88where:integralsignf(x):integranda:lowerlimitofintegrationb:upperlimitofintegrationTheprocedureofcalculatinganintegraliscalledintegration.where:integralsignf(x):int89Caution:Thedefiniteintegralisanumber;itdoesnotdependonx.Infact,wecoulduseanyletterinplaceofxwithoutchangingthevalueoftheintegral:wherexisadummyvariable.Caution:Thedefiniteintegral90Geometricinterpretation::theareaoftheregion:thenegativeoftheareaoftheregion:thealgebraicsumoftheorientedarea,orthenetarea.Geometricinterpretation::the91Theorem1.Theorem2.andhaveonlyfinitediscontinuitieson[a,b]Thesufficientconditionofintegrability:isintegrableon

isintegrableon

Theorem1.Theorem2.andhaveonl92Example1.UsethedefinitiontoevaluateSolution:UseWechoosetodivide[0,1]intonsubintervalsofequalwidth.Example1.Usethedefinition93微積分第五章課件94Example2.UsethedefinitiontoevaluateSolution:UseWechoosetodivide[1,3]intonsubintervalsofequalwidth.Example2.Usethedefinition95微積分第五章課件96Solution:Usetheintegraltodenotethefollowinglimits:Solution:Usetheintegraltod97Usethegeometricmeaningtoevaluate:Usethegeometricmeaningtoe98TheMidpointRuleTheMidpointRule99PropertiesofthedefiniteintegralWhenwedefinedthedefiniteintegral.weimplicitlyassumedthata<b.ButthedefinitionasalimitofRiemannsumsmakessenseevenifa>b.Noticethatifwereverseaandb,thenchange(b-a)/nfromto(a-b)/n.ThereforePropertiesofthedefiniteint100微積分第五章課件101Propertiesoftheintegral(kisaconstant)Proof:(cisaconstant)Propertiesoftheintegral(k102

Example:Usethepropertiesofintegralstoevaluate

Example:Usethepropert103微積分第五章課件104ComparisonPropertiesoftheintegral:ComparisonPropertiesofthei105ThenexttheoremiscalledMeanValueTheoremfor

DefiniteIntegrals.Itsgeometricinterpretationisthat,fora

continuouspositivef(x)on[a,b],thereisanumbercin[a,b]suchthattherectanglewithbase[a,b]andheightf(c)hasthesameareaastheregionunderthegraphoff(x)fromatob.oabxycf(x)TheoremIff(x)iscontinuouson[a,b],thenthereexistsatleastanumbercin(a,b)suchthatThenexttheoremiscalledMea106Thenumberiscalledtheaveragevalueoff(x)on[a,b].Proof

Iff(x)isaconstantfunction,theresultistrue.Nextweassumethatf(x)isnotaconstantfunction.Sincef(x)iscontinuouson[a,b],f(x)takesontheminimumandthemaximumvalueson[a,b].Letf(u)=mandf(v)=Mbetheminimumandthemaximumvaluesoff(x)on[a,b],respectively.Thenm<f(x)<Mforsomexin[a,b]becausef(x)isnotconstant.Therefore,wehaveItfollowsthat

Thenumber107Theprecedinginequalitiesindicatethatthenumber

isbetweenm=f(u)andM=f(v).ThustheIntermediateValueTheoremleadsthatthereisanumbercbetweenuandvsuchthatMultiplyingthebothsidesbyb-agivestheconclusionofthetheorem.Theprecedinginequalitiesin1085.3TheFundamentalTheorem

ofCalculus5.3TheFundamentalTheorem109TheFundamentalTheoremofCalculusThefundamentalTheoremofCalculusisappropriatelynamedbecauseitestablishesaconnectionbetweenthetwobranchestoCalculus:differentialcalculusandintegralcalculus.Differentialcalculusarosefromthetangentproblem,whereasintegralcalculusarosefromaseeminglyunrelatedproblem,theareaproblem.TheFundamentalTheoremofCalculusgivesthepreciserelationshipbetweenthederivativeandtheintegral.TheFundamentalTheoremofCa110Newton’steacheratCambridge,IssacBarrowdiscoveredthatthetwoproblemareactuallycloselyrelated.Infact,herealizedthatdifferentiationandintegrationareinverseprocesses.ItwasNewtonandLeibnizwhoexploitedthispreciserelationshipanduseittodevelopcalculusintoasystematicmathematicalmethod.Inparticular,theysawthattheFundamentalTheoremenabledthemtocomputeareasandintegralsveryeasilywithouthavingtocomputethemaslimitsofsums.Newton’steacheratCambridge,111ThefirstpartoftheFundamentalTheoremdealswithfunctionsdefinedbyanequationoftheformwherefisacontinuousfunctionon[a,b]andxvariesbetweenaandb.Observethatgdependsonlyonx,whichappearsasthevariableupperlimitintheintegral.Ifxisafixednumber,thentheintegralisadefinitenumber.Ifwethenletxvary,thenumberalsovariesanddefinesafunctionofxbyg(x).baxf(t)ThefirstpartoftheFundamen112TheFundamentalTheoremofCalculus,PartIProof:thenTheFundamentalTheoremofCal113微積分第五章課件114微積分第五章課件115微積分第五章課件116TheFundamentalTheoremofCalculus,Part2Proof:Accordingtopart1,sosowehavedenoteTheFundamentalTheoremofCal117微積分第五章課件118微積分第五章課件119Why?Why?120Summary:WeendthissectionbybringingtogetherthetwopartsoftheFundamentalTheorem.Summary:Weendthissectionby1215.4IndefiniteIntegrals

andtheNetChangeTheorem5.4IndefiniteIntegrals122IndefiniteIntegrals

andtheNetChangeTheoremAnindefiniteintegraloff(x)representanentirelyfamilyoffunctionswhosederivativeisf(x)

andisdenotedbySupposethatF(x)isanantiderivativeoff(x),thenaccordingtothetheoremin4.2,weknowthatanyantiderivativeG(x)off(x)canbewrittenas

G(x)=F(x)+CIndefiniteIntegrals

and123Then:Caution:1)Youshoulddistinguishcarefullybetweendefiniteandindefiniteintegrals.Adefiniteintegralisanumber,whereasanindefiniteintegralisafunction(afamilyoffunctions).2)Theconnectionisthefundamentaltheorem:Then:Caution:124TableofIndefiniteintegralsFromtheabovedefinition,weknownthat:oror(kisaconstant)TableofIndefiniteintegrals125orororor126微積分第五章課件127Example1:Sulotion:=Example2:Solution:=Example1:Sulotion:=128ThepropertiesofIndefiniteIntegralsCorollary:IfthenThepropertiesofIndefiniteI129Example3:Solution:=Example3:Solution:=130Example4:Solution:=Example5:Solution:=Example4:Solution:=Example5131Example:

Solution:=Example:Solution