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Chapter15MultipleIntegrals15.1DoubleIntegralsoverRectangles15.2IteratedIntegrals15.3DoubleIntegralsoverGeneralRegions15.4DoubleIntegralsinpolarcoordinates15.5*ApplicationsofDoubleIntegrals15.6*SurfaceArea,15.1DoubleIntegralsoverRectangles,VolumesandDoubleIntegrals,Afunctionfoftwovariablesdefinedonaclosedrectangle,andwesupposethat,Thegraphoffisasurfacewithequation,LetSbethesolidthatliesaboveRandunderthegraphoff,thatis,(SeeFigure1)FindthevolumeofS,Figure1,1)Partition:,ThefirststepistodividetherectangleRintosubrectangles.,Eachwitharea,2)Approximation:,Athinrectangularbox:,Base:,Height:,Wecanapproximateby,3)Sum:,4)Limit:,AdoubleRiemannsum,DefinitionThedoubleintegraloffovertherectangleRis,ifthislimitexists.,Thesufficientconditionofintegrability:,isintegralonR,Theorem1.,Theorem2.,andfisdiscontinuousonlyonafinitenumberofsmoothcurves,isintegralon,Note,IfthenthevolumeVofthesolidthat,liesabovethatthesurfaceis,Example1,If,evaluatetheintegral,Solution,15.2IteratedIntegrals,Partialintegrationwithrespecttoydefinesafunctionofx:,WeintegrateAwithrespecttoxfromx=atox=b,weget,Theintegralontherightsideiscalledaniteratedintegralandisdenotedby,Thus,Similarly,FubinistheoremIffiscontinuousontherectangle,then,Moregenerally,thisistruethatweassumethatfisboundedonR,fisdiscontinuousonlyonafinitenumberofsmoothcurves,andtheiteratedintegralsexist.,TheproofofFubinistheoremistoodifficulttoincludeInourclass.,Iff(x,y)0,thenwecaninterpretthedoubleintegral,asthevolumeVofthesolidSthatliesabove,Randunderthesurfacez=f(x,y).,So,Or,Example,Solution,Example,Solution,Example,Solution,Specially,If,Then,SomeexamplesoftypeI,SomeexamplesoftypeII,Example4,Findthevolumeofthesolidenclosed,bytheparaboloidandtheplanes,Solution,andaboveregion,Thesolidliesundertheparaboloid,Sothevolumeis,SupposethatDisaboundedregion,thedoubleintegraloffoverDis,15.3DoubleIntegralsoverGeneralRegions,SupposethatDisaboundedregionwhichcanbeenclosedinarectangularregionR.,AnewfunctionFwithdomainR:,IftheintegralofFexistsoverR,thenwedefinethedoubleintegraloffoverDby,SomeexamplesoftypeI,EvaluatewhereDisaregionoftypeI,AnewfunctionFwithdomainR:,IffiscontinuousontypeIregionDsuchthat,then,SomeexamplesoftypeII,IffiscontinuousontypeIIregionDsuchthat,then,Example1,Solution,TypeI,TypeII,PropertiesofDoubleIntegralSupposethatfunctionsfandgarecontinuousonaboundedclosedregionD.Property1Thedoubleintegralofthesum(ordifference)oftwofunctionsexistsandisequaltothesum(ordifference)oftheirdoubleintegrals,thatis,Property2Property3whereDisdividedintotworegionsD1andD2andtheareaofD1D2is0.,Property4Iff(x,y)0forevery(x,y)D,thenProperty5Iff(x,y)g(x,y)forevery(x,y)D,thenMoreover,sinceitfollowsfromProperty5thathence,whereSistheareaofD.,Property6,Property7SupposethatMandmarerespectivelythemaximumandminimumvaluesoffunctionfonD,thenwhereSistheareaofD.Property8(TheMeanValueTheoremforDoubleIntegral)Iff(x,y)iscontinuousonD,thenthereexistsatleastapoint(,)inDsuchthatwhereSistheareaofD.f(,)iscalledtheaverageValueoffonD,Example2,Solution,TypeII,TypeI,Example3,Solution,TypeI,TypeII,Examplechangetheorderofintegration,solution：,Wehave,AnalternativedescriptionofDis,Examplechangetheorderofintegration,solution：,ExampleProvethat,Solution,AnalternativedescriptionofDis,where,Wehave,So,Chapter10ParametricEquationsandPolarCoordinates10.3Polarcoordinates,10.3Polarcoordinates,ThepointoiscalledthepoleThepointPisrepresentedbytheorderedpair(r,)andr,arecall-edpolarcoordinatesofPispositiveifmeasureinthecou-nterclockwisedirectionfromthepo-laraxisandnegativeintheclockwi-sedirection,TheconnectionbetweenpolarandCartesiancoordinates,ExampleConvertthepointfrompolartoCartesiancoordinates,Solution,ExampleRepresentthepointwithCartesiancoordinates,Solution,intermsofpolarcoordinates.,ExampleIdentifythecurvebyfindingaCartesianequationforthecurve,Solution,ExampleFindapolarequationforthecurverepresentedbythegivenCartesianequation,Solution,Chapter15MultipleIntegrals15.1DoubleIntegralsoverRectangles15.2IteratedIntegrals15.3DoubleIntegralsoverGeneralRegions15.4DoubleIntegralsinpolarcoordinates15.5*ApplicationsofDoubleIntegrals15.6*SurfaceArea,15.4DoubleIntegralsinpolarcoordinates,Apolarrectangle,where,The“center”ofthepolarsubrectangle,haspolarcoordinates,Theareaofis,Changetopolarcoordinateinadoubleintegral,IffiscontinuousonapolarrectangleRgivenby,where,then,Solution,1.Iffiscontinuousonapolarregionoftheformthen,2.Iffiscontinuousonapolarregionoftheformthen,3.Iffiscontinuousonapolarregionoftheformthen,Solution：,ExampleFind，。,Disgivenby,So,ExampleEvaluate，,Solution,Wehave,ImproperIntegral(overtheentireplane),whereisthediskwithradiusandcentertheorigin.,whereisthesquarewithvertics.,Example,15.7TripleIntegrals,fisdefindontherectanglarboxB,DefinitionThetripleintegraloffovertheboxBis,ifthislimitexists.,isintegralonB,Theorem1.,FubinistheoremIffiscontinuousontherectanglar,then,boxB,Example,Evaluatethetripleintegral,WhereBistherectangularboxgivenby,Solution,ThetripleintegraloverageneralboundedregionEinthree-dimensionalspace,AsolidregionEissaidtobeoftypeIif,then,IftheprojectionDofEontothexy-planeisatypeIplaneregion,then,then,IftheprojectionDofEontothexy-planeisatypeIIplaneregion,Example,Evaluate,whereEisthesolid,tetrahedronboundedbythefourplanes,and,tetrhi:drn,Solution,Wehave,itisatypeIregion.,15.8TripleIntegralsinCylindricalandSphericalCoodinates,1.CylindricalCoodinates,If,whereDisgiveninpolarcoordinatesby,then,Example,AsolidElieswithinthecylinder,belowtheplane,Solution,abovetheparaboloid,Thedensityatanypointisproportional,toitsdistancefromtheaxisofthecylinder.,FindthemassofE.,Wehave,Sincethedensityatanypointisproportional,toitsdistancefromthez-axis,thedensityfunctionis,wherekistheproportionalityconstant.Therefore,themassis,Example,Evaluatetheintegralbychangingtocylindricalcoordinates.,Solution,Thesolidregionhasamuchsimplerdescriptionincylindricalcoordinates:,Thesolidregionis,2.SphericalCoodinates,whereEisasphericalwedgegivenby,IfEisageneralsphericalregionsuchas,then,Example,Evaluatetheintegralbychangingtosphericalcoordinates.,Solution,Thesolidregionhasamuchsimplerdescriptioninsphericalcoordinates:,Thesolidregionis,Example,Usesphericalcoordinatestofindthevolume,Solution,ThevolumeofEis,Thesolidis,ofthesolidthatliesabovetheconeand,belowthesphere,15.9ChangeofVariablesinMutipleIntegrals,AtransformationTfromtheuv-planetothexy-plane,where,or,Tisatransformation,whichmeansthatandhavecontinuousfirst-orderpartialderivatives.,AtransformationTisafunctionwhosedomanandrangearebothsubsetsofIfthenthepointiscalledtheimageofthepointIfnotwopointshavethesameimage,Tiscalledone-to-one.,AtransformationTonaregionSintheuv-plane.,TtransformsSintoaregionRinthexy-planecalledtheimageofS,consistingoftheimagesofallpointsinS.,IfTisone-to-onetransformation,thenithasaninversetransformationfromthexy-planetotheuv-plane.,Example,Atransformationisdefinedby,Sisthetriangularregionwithvertices,Solution,Thetransformatonmapstheboundaryof,Si