上海财经大学英语高数课件05

Chapter5Integrals机动目录上页下页返回结束5.2Area5.3TheDefiniteIntegral5.4TheFundamentalTheoremofCalculus5.5TheSubstitutionRule4.2AreaAreaProblem: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Thenthesumoftheareaoftheserectanglesis26)12)(1(21221211)()()(nnnnnnnnnnnSAsnincreases,Snbecomesabetterandbetterapproximationtotheareaoftheparabolicsegment.ThereforewedefinetheareaAtobethelimitofthesumsoftheareasoftheserectangles,thatis,31limnnSAApplyingtheideaofExample1tothemoregeneralregionSofF.1,weintroducethedefinitionoftheareaasfollowing:Step1:Partition--Dividetheinterval[a,b]intonsmallersubintervalsbychoosingpartitionpointsx0,x1,x2,….,xnsothata=x0x1x2…xn=b机动目录上页下页返回结束Step2:Approximation—Bythepartitionabove,theareaofScanbeapproximatedbythesumofareasofnrectangles.Thissubdivisioniscalledapartitionof[a,b]andwedenoteitbyP.Letdenotethelengthofithsubinterval[xi-1,xi],and||P||(thenormofP)denotesthelengthoflongestsubinterval.Thus1iiixxx},,max{||||1nxxPUsingthepartitionPonecandividetheregionSintonstrips(seeF.3).Now,wechooseanumberineachsubinterval,theneachstripSicanbeapproximatedbyarectangleRi(seeF.4).ixThesumofareasoftheserectanglesasanapproximationisniiiniixxfA11)(机动目录上页下页返回结束y=f(x)SiabxyoFigure3XiXi-1S1S2SnR2y=f(x)RiabxyoFigure4XiXi-1R1Rnix1x2xnxapproximatedby机动目录上页下页返回结束Step3:Takinglimit—Noticethattheapproximationappearstobecomebetterandbetterasthestripsbecomethinnerandthinner.Sowedefinetheareaoftheregionasthelimitvalue(ifitexists)ofthesumofareasoftheapproximatingrectangles,thatisniiiPxxfA10||||)(lim(1)Remark2:Instep1,wehavenoneedtodividedtheinterval[a,b]intonsubintervalswithequallength.Butforpurposesofcalculation,itisoftenconvenienttotakeapartitionthatdividestheintervalintonsubintervalswithequallength.(Thisiscalledaregularpartition)Remark1:Itcanbeshownthatiffiscontinuous,thenthelimit(1)doesexist.机动目录上页下页返回结束SolutionSincey=x2+1iscontinuous,thelimit(1)mustexistforallpossiblepartitionPoftheinterval[a,b]aslongas||P||0.Tosimplifythingsletustakearegularpartition.Thenthepartitionpointsarex0=0,x1=2/n,x2=4/n,…,xi=2i/n,…,xn=2n/n=2SothenormofPis||P||=2/nLetuschoosethepointtobetheright-handendpoint:=xi=2i/nBydefinition,theareaisixExample2:Findtheareaundertheparabolay=x2+1from0to2.ix机动目录上页下页返回结束Example3Findtheareaunderthecosinecurvefrom0tob,where.2/0bSolutionWechoosearegularpartitionPsothat||P||=b/nandwechoosetobetheright-handendpointoftheithsub-interval:=xi=ib/nSince||P||0asn,theareaunderthecosinecurvefrom0tobisixix31412210||||)(lim)(limninninniiiPfxxfARemark3:Ifischosentobetheleft-handendpoint,onewillobtainthesameanswer.ix机动目录上页下页返回结束bxxfAnbnbnbnbnninibnbnniiiPsinlimcoslim)(lim22)1(2sincossin110||||/section5.2end机动目录上页下页返回结束5.3TheDefiniteIntegralInChapters6and8wewillseethatlimitofformniiiPxxfA10||||)(limoccursinawidevarietyofsituationsnotonlyinmathematicsbutalsoinphysics,Chemistry,BiologyandEconomics.Soitisnecessarytogivethistypeoflimitaspecialnameandnotation.1.DefinitionofaDefiniteIntegralIffisafunctiondefinedonaclosedinterval[a,b],letPbeapartitionof[a,b]withpartitionpointsx0,x1,x2,….,xn,wherea=x0x1x2…xn=b机动目录上页下页返回结束Choosepointsin[xi-1,xi]andletand1iiixxx}max{||||ixPix.ThenthedefiniteintegralofffromatobisniiiPbaxxfdxxf10||||)(lim)(ifthislimitexists.Ifthelimitdoesexist,thenfiscalledintegrableontheinterval[a,b].upperlimitintegrandNote1:badxxf)(lowerlimitintegralsign机动目录上页下页返回结束Note2:isanumber;itdoesnotdependonx.Infact,wehavebadxxf)(bababadrrfdttfdxxf)()()(Note3:ThesumisusuallycalledaRiemannsum.niiixxf1)(Note4:GeometricinterpretationsForthespecialcasewheref(x)0,=theareaunderthegraphofffromatob.badxxf)(Ingeneral,adefiniteintegralcanbeinterpretedasadifferenceofareas:21)(AAdxxfbaxyoab++-Note5:Inthecaseofabanda=b,weextendthedefinitionofasfollows:badxxf)(Ifab,thenabbadxxfdxxf)()(Ifa=b,then0)(aadxxfExample1ExpressniiiiiPxxxx130||||]sin)[(limasanintegralontheinterval[0,].Example2Evaluatetheintegralbyinterpretingintermsofareas.30)1(dxx机动目录上页下页返回结束SolutionWecomputetheintegralasthedifferenceoftheareasofthetwotriangles:5.1)1(2130AAdxx-113oxyy=x-1A1A22.ExistenceTheoremTheoremIffiseithercontinuousormonotonicon[a,b],thenfisintegrableon[a,b];thatis,thedefiniteintegralexists.badxxf)(Remark1:Iffisdiscontinuousatsomepoints,thenmightexistoritmightnotexist.Butiffispiecewisecontinuous,thenfisintegrable.badxxf)(机动目录上页下页返回结束Remark2:Itcanbeshownthatiffisintegrableon[a,b],thenfmustbeaboundedfunctionon[a,b].3.IntegralFormulasunderRegularPartitionIffisintegrableon[a,b],itisoften