高等数学英文版课件PPT 05 Integrals.pptx

高等数学英文版课件PPT 05 Integrals.pptx

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Chapter5Integrals2.Area3.TheDefiniteIntegral4.TheFundamentalTheoremofCalculus5.TheSubstitutionRule机动目录上页下页返回结束4.2AreaAreaProblem:

FindtheareaoftheregionSthatliesunderthecurvey=f（x）fromatob.（seeFigure1）y=f（x）SbxyoaFigure1机动目录上页下页返回结束Ideaforproblemsolving:

FirstapproximatetheregionSbypolygons,andthentakethelimitoftheareaofthesepolygons.（seethefollowingexample）Example1Findtheareaundertheparabolay=x2from0to1.ySolution:

Westartbydividingtheinterval0,1inton-subintervalswithequallength,andconsidertherectangleswhosebasesarethesesubintervalsandwhoseheightsarethevaluesofthefunctionattheright-handendpoints.o（1,1）1/n1xFigure2机动目录上页下页返回结束ThenthesumoftheareaoftheserectanglesisAsnincreases,Snbecomesabetterandbetterapproximationtotheareaoftheparabolicsegment.ThereforewedefinetheareaAtobethelimitofthesumsoftheareasoftheserectangles,thatis,ApplyingtheideaofExample1tothemoregeneralregionSofF.1,weintroducethedefinitionoftheareaasfollowing:

Step1:

Partition-Dividetheintervala,bintonsmallersubintervalsbychoosingpartitionpointsx0,x1,x2,.,xnsothata=x0x1x2xn=b机动目录上页下页返回结束Thissubdivisioniscalledapartitionofa,bandwedenoteitbyP.Letdenotethelengthofithsubintervalxi-1,xi,and|P|（thenormofP）denotesthelengthoflongestsubinterval.ThusStep2:

ApproximationBythepartitionabove,theareaofScanbeapproximatedbythesumofareasofnrectangles.UsingthepartitionPonecandividetheregionSintonstrips（seeF.3）.Now,wechooseanumberineachsubinterval,theneachstripSicanbeapproximatedbyarectangleRi（seeF.4）.Thesumofareasoftheserectanglesasanapproximationis机动目录上页下页返回结束y=f（x）SibxyoaFigure3XiXi-1S1S2SnR2Ribxyy=f（x）oaFigure4XiXi-1R1Rnapproximatedby机动目录上页下页返回结束Step3:

TakinglimitNoticethattheapproximationappearstobecomebetterandbetterasthestripsbecomethinnerandthinner.Sowedefinetheareaoftheregionasthelimitvalue（ifitexists）ofthesumofareasoftheapproximatingrectangles,thatis

（1）Remark1:

Itcanbeshownthatiffiscontinuous,thenthelimit

（1）doesexist.Remark2:

Instep1,wehavenoneedtodividedtheintervala,bintonsubintervalswithequallength.Butforpurposesofcalculation,itisoftenconvenienttotakeapartitionthatdividestheintervalintonsubintervalswithequallength.（Thisiscalledaregularpartition）机动目录上页下页返回结束tobetheright-handendpoint:

Letuschoosethepoint=xi=2i/nBydefinition,theareaisExample2:

Findtheareaundertheparabolay=x2+1from0to2.SolutionSincey=x2+1iscontinuous,thelimit

（1）mustexistforallpossiblepartitionPoftheintervala,baslongas|P|0.Tosimplifythingsletustakearegularpartition.Thenthepartitionpointsarex0=0,x1=2/n,x2=4/n,xi=2i/n,xn=2n/n=2SothenormofPis|P|=2/n机动目录上页下页返回结束Remark3:

Ifischosentobetheleft-handendpoint,onewillobtainthesameanswer.Example3Findtheareaunderthecosinecurvefrom0tob,whereSolutionWechoosearegularpartitionPsothat|P|=b/nandwechoosetobetheright-handendpointoftheithsub-interval:

=xi=ib/nSince|P|0asn,theareaunderthecosinecurvefrom0tobis机动目录上页下页返回结束/section5.2end机动目录上页下页返回结束5.3TheDefiniteIntegralInChapters6and8wewillseethatlimitofformoccursinawidevarietyofsituationsnotonlyinmathematicsbutalsoinphysics,Chemistry,BiologyandEconomics.Soitisnecessarytogivethistypeoflimitaspecialnameandnotation.1.DefinitionofaDefiniteIntegralIffisafunctiondefinedonaclosedintervala,b,letPbeapartitionofa,bwithpartitionpointsx0,x1,x2,.,xn,wherea=x0x1x20,=theareaunderthegraphofffromatob.Ingeneral,adefiniteintegralcanbeinterpretedasadifferenceofareas:

xyob+a-ofNote5:

Inthecaseofabanda=b,weextendthedefinitionasfollows:

Ifab,thenIfa=b,thenExample1Expressbyinterpretinginasanintegralontheinterval0,.Example2Evaluatetheintegraltermsofareas.机动目录上页下页返回结束SolutionWecomputetheintegralasthedifferenceoftheareasofthetwotriangles:

13o-1xyy=x-1A1A22.ExistenceTheoremTheoremIffiseithercontinuousormonotonicona,b,thenfisintegrableona,b;thatis,thedefiniteintegralexists.Remark1:

Iffisdiscontinuousatsomepoints,thenmightexistoritmightnotexist.Butiffispiecewisecontinuous,thenfisintegrable.机动目录上页下页返回结束Remark2:

Itcanbeshownthatiffisintegrableona,b,thenfmustbeaboundedfunctionona,b.3.IntegralFormulasunderRegularPartitionIffisintegrableona,b,itisoftenconvenienttotakearegularpartition.Thenandtobetherightendpointineachsubinterval,Ifwechoosethen0asn,sothedefinitionSince|P|=（b-a）/n,wehave|P|gives机动目录上页下页返回结束TheoremIffisintegrableona,b,thenasanintegralontheExample3Expressinterval1,2.Answer:

Ifthepurposeistofindanapproximationtoanintegral,itisusuallybettertochoosewhic