高等数学英文版课件PPT 05 Integrals.ppt 37页

'Chapter5Integrals机动目录上页下页返回结束5.2Area5.3TheDefiniteIntegral5.4TheFundamentalTheoremofCalculus5.5TheSubstitutionRule 4.2AreaAreaProblem:FindtheareaoftheregionSthatliesunderthecurvey=f(x)fromatob.(seeFigure1)y=f(x)SabxyoFigure1机动目录上页下页返回结束 Ideaforproblemsolving:FirstapproximatetheregionSbypolygons,andthentakethelimitoftheareaofthesepolygons.(seethefollowingexample)Example1Findtheareaundertheparabolay=x2from0to1.ySolution:Westartbydividingtheinterval[0,1]inton-subintervalswithequallength,andconsidertherectangleswhosebasesarethesesubintervalsandwhoseheightsarethevaluesofthefunctionattheright-handendpoints.ox(1,1)1/n1Figure2机动目录上页下页返回结束 ThenthesumoftheareaoftheserectanglesisAsnincreases,Snbecomesabetterandbetterapproximationtotheareaoftheparabolicsegment.ThereforewedefinetheareaAtobethelimitofthesumsoftheareasoftheserectangles,thatis,ApplyingtheideaofExample1tothemoregeneralregionSofF.1,weintroducethedefinitionoftheareaasfollowing:Step1:Partition--Dividetheinterval[a,b]intonsmallersubintervalsbychoosingpartitionpointsx0,x1,x2,….,xnsothata=x00,=theareaunderthegraphofffromatob.Ingeneral,adefiniteintegralcanbeinterpretedasadifferenceofareas:xyoab++- Note5:Inthecaseofa>banda=b,weextendthedefinitionofasfollows:Ifa>b,thenIfa=b,thenExample1Expressasanintegralontheinterval[0,].Example2Evaluatetheintegralbyinterpretingintermsofareas.机动目录上页下页返回结束 SolutionWecomputetheintegralasthedifferenceoftheareasofthetwotriangles:-113oxyy=x-1A1A22.ExistenceTheoremTheoremIffiseithercontinuousormonotonicon[a,b],thenfisintegrableon[a,b];thatis,thedefiniteintegralexists.Remark1:Iffisdiscontinuousatsomepoints,thenmightexistoritmightnotexist.Butiffispiecewisecontinuous,thenfisintegrable.机动目录上页下页返回结束 Remark2:Itcanbeshownthatiffisintegrableon[a,b],thenfmustbeaboundedfunctionon[a,b].3.IntegralFormulasunderRegularPartitionIffisintegrableon[a,b],itisoftenconvenienttotakearegularpartition.ThenandIfwechoosetobetherightendpointineachsubinterval,thenSince||P||=(b-a)/n,wehave||P||0asn,sothedefinitiongives机动目录上页下页返回结束 TheoremIffisintegrableon[a,b],thenExample3Expressasanintegralontheinterval[1,2].Answer:Ifthepurposeistofindanapproximationtoanintegral,itisusuallybettertochoosetobemidpointofthesubinterval,whichwedenoteby.AnyRiemannsumisanapproximationtoanintegral,butifweusemidpointsandaregularpartitionwegetthefollowingapproximation:机动目录上页下页返回结束 MidpointRulewhereandUsingtheMidpointRulewithn=5wecangetanapproximationofintegral(seepage277).机动目录上页下页返回结束 4.PropertiesoftheIntegralSupposeallofthefollowingintegralsexist.ThenExample4Usingthepropertiesaboveandtheresults机动目录上页下页返回结束toevaluate OrderpropertiesoftheintegralSupposethefollowingintegralsexistanda0andag(x)fora0,wehaveThus,byProperty7, 4.4TheFundamentalTheoremofCalculusTheFundamentalTheoremofCalculusinthissectiongivesthepreciseinverserelationshipbetweenthederivativeandtheintegral.Itenablesustocomputeareasandintegralsveryeasilywithouthavingtocomputethemaslimitsofsumsaswedidinsections4.2and4.3.1.FundamentalTheoremForacontinuousfunctionfon[a,b],wedefineanewfunctiongbyComputingthederivativeofg(x)weobtain机动目录上页下页返回结束 TheFundamentalTheoremofCalculus,Part1Iffiscontinuouson[a,b],thenthefunctiondefinedbyiscontinuouson[a,b]anddifferentiateon(a,b),andProofIfxandx+harein[a,b],thenandso,forh,(2)机动目录上页下页返回结束 Fornowletusassumethath>0.Sincefiscontinuouson[x,x+h],theExtremeValueTheoremsaysthattherearenumberuandvin[x,x+h]suchthatf(u)=mandf(v)=M,wheremandMaretheabsoluteminimumandmaximumvaluesoffon[x,x+h].机动目录上页下页返回结束Inequality(3)canbeprovedinasimilarmannerforthecasewhereh<0.(3)Combiningitwith(2)givesSinceh>0,wecandividethisinequalitybyh:ByProperty7,wehave Sincefiscontinuousatx,andu,vliebetweenxandx+h,wehave机动目录上页下页返回结束Ifx=aandb,thenEquation(4)canbeinterpretedasone-sidelimit.ThenTheorem2.1.8showsthatgiscontinuouson[a,b].(4)Weconclude,from(3)andtheSqueezeTheorem,that Note:ThisTheoremcanbewritteninLeibniznotationas机动目录上页下页返回结束Roughlyspeaking,equation(5)saysthatwefirstintegratefandthendifferentiatetheresult,wegetbacktotheoriginalfunctionf.(5) Example1Findthederivativeofthefunctiong(x)=Example2FindExample3FindExample4FindThesecondpartoftheFundamentalTheoremofCalculusprovidesuswithamuchsimplermethodfortheevaluationofintegrals.TheFundamentalTheoremofCalculus,Part2Iffiscontinuouson[a,b],then机动目录上页下页返回结束whereFisanyantiderivativeoff,thatis ProofWeknowfrompart1thatgisanantidervitiveoff.IfFisanyotherantidervitiveoffon[a,b],thenfandgdifferonlybyaconstant:(6)F(x)=g(x)+Cfora