《高数双语》课件section 10.2

Section10.2EvaluationofDoubleIntegrals1Riemann,Bernhard2Geometricmeaningofthedoubleintegralcylindricalbodyinthreedimensionalspace.VolumeofaCylindricalbodyσthenitcanbethinkofaSupposethat3VolumeofaCylindricalbodyVolume=4CalculatingDoubleIntegralsOverRectanglesiscontinuousdefinedonarectangularregion(σ)ThenwemakeanetworkoflinesparallelTheselinesdivide(σ)intotox-andy-axes.WenumbersmallpiecesofareatheseinsomeorderthenChooseapointineachpiecegivenby5gotozero,thesumsapproachalimitcalledtheCalculatingDoubleIntegralsOverRectanglesIffiscontinuousthroughout(σ),thenaswerefinedthemeshwidthtomakebothandThenotationforitisdoubleintegraloffover(σ).orThus,Aswithfunctionsofasinglevariable,thesumsapproachthislimitnothatdetermine(σ)arepartitioned,matterhowtheintervalsandaslongasthenormsofthepartitionsbothgotozero.6Fubini’sTheoremforCalculatingDoubleIntegralsSupposethatwewishtocalculatethevolumeundertheplaneonthexy-plane.overtherectangularregionIfwedenotetheareaofthecross-sectionatxasForeachvalueofx,wemaycalculate,thenthevolumeisastheintegralwhichistheareaunderthecurveintheplaneofthecrosssectionatx.7Fubini’sTheoremforCalculatingDoubleIntegrals,xisheldfixedandtheintegrationtakesplacewithIncalculatingThenthevolumeoftheentiresolidisrespecttoy.Ifwejustwanttowriteinstructionsforcalculatingthevolume,withoutcarryingoutanyoftheintegrations,wecouldhavewritten8Fubini’sTheoremforCalculatingDoubleIntegralsWhatwouldhavehappenedifwehadcalculatedthevolumebyslicingwithplanesperpendiculartothey–axis?Asafunctionofy,thetypicalcross-sectionareaisTherefore9Fubini’sTheoremforCalculatingDoubleIntegralsTheoremFubini’sTheorem(FirstForm)iscontinuousthroughouttherectangularIfthenregionFubini’stheoremsaysthatdoubleintegralsoverrectanglescanbeThus,wecanevaluatecalculatedasiteratedintegrals[累次积分].adoubleintegralbyintegratingwithrespecttoonevariableatatime.10EvaluatingaDoubleIntegralExampleforCalculateandSolutionByFubini’stheorem,Reversingtheorderofintegrationgivesthesameanswer:11DoubleIntegralsoverBoundedNonrectangularRegionsIf

(Ω)

isaregionliketheoneshowninthexy-planebounded“above”and“below”bytheintheleftfigure,andonthesidescurvesandwemayagainbythelinescalculatethevolumebythemethodWefirstcalculatethecross-sectionareaofslicing.iscontinuousdefinedonaclosedregion(Ω).12DoubleIntegralsoverBoundedNonrectangularRegionsfromtogetthevolumeasanThentheintegratetoiteratedintegral13DoubleIntegralsoverBoundedNonrectangularRegions,thentheSimilarly,if

(Ω)

isaregionliketheoneshownintherightfigure,boundedbytheandcurvesandthelinesandvolumecalculatedbyslicingisgivenbytheiteratedintegral14DoubleIntegralsoverBoundedNonrectangularRegionsTheoremFubini’sTheorem(StrongerFrom)becontinuousonaregion(Ω).Letwithh1andh2continuous2.If(Ω)isdefinedby,thenon1.If(Ω)isdefinedbywithg1andg2continuous,thenony–typeregionx–typeregionDoubleIntegralsoverBoundedNonrectangularRegions15Example

Findwhere(Ω)istheregionboundedbythelinex=1,y=0andtheparabolax=1

OxyDoubleIntegralsoverBoundedNonrectangularRegions16Example

Findwhere(Ω)istheregionboundedbythelinex=1,y=0andtheparabolax=1

Oxy117FindingVolumeExampleFindthevolumeoftheprismwhoseandandwhosetopliesintheplanebaseisthetrianglein

thexy–planeboundedbythex–axisandthelinesSolutionForanyxbetween0and1,ymayHence,mayvaryfromto18FindingVolumeSolution(continued)Whentheorderofintegrationisreversed,theintegralforthevolumeisThetwointegralsareequal,astheyshouldbe.19DoubleIntegralsoverBoundedNonrectangularRegionsNoteabxyOcdΩx0y0Iftheregion(Ω)

isofbothx-typeandy-type,thenfromtheFubini’sTheorem(StrongerFrom)weknow20ReversingTheOrderofIntegrationExample

Sketchtheregionofintegrationfortheintegralandwriteanequivalentintegralwiththeorderofintegrationreversed.SolutiontheregionboundedbythecurvesandTherefore,betweenandTofindtheintegratinginthereverseorder,weimagineahorizontallinepassingfromlefttorightthroughtheregion.21ReversingTheOrderofIntegrationSolution(continued)andleavesatThecommonvalueoftheseintegralsis8.Toincludeallsuchlines,welety

ThelineentersatTheintegralisrunfromto22FindingAreaExample

Findtheareaoftheregion(Ω)

enclosedbytheparabolaandthelineSolutionIfwedivide(Ω)

intotheregionsΩ

1andΩ

2shownintherightfigure,wemaycalculatetheareaas23FindingAreaSolution(continued)Ontheotherhand,reversingtheorderofintegrationgivesThisresultissimplerandistheonlyonewewouldbothertowritedowninTheareaispractice.Example

Findtheareaoftheregion(Ω)enclosedbytheparabolaandtheline24IntegralsinPolarCoordinatesisdefinedoveraregion(Ω)thatisboundedSupposethatafunctionandbythecontinuouscurvesbytheraysandforeveryvalueandSupposealsothatThen(Ω)liesinafan-shapedregionQdefinedofφ

betweenα

andβ.andbytheinequalitiesapproachalimitaswerefinethegridtomakeIntegralsinPolarCoordinatesbethecenterofthepolarrectanglewhoseareaisWeletBy“center”,wemeanthepointthatlieshalfwaybetweenthecircularWethenformthesumarcontheraythatbisectsthearcs.IffiscontinuousthroughoutΩ,thissumwillLargesectorSmallsectorgotozero.andTheareasofthecircularsectorssubtendedbythesearcsattheoriginareOuter:Inner:26IntegralsinPolarCoordinatesTherefore,Then,BytheFubini’stheoremnowsaysthatthelimitapproachedbythesesumscanbeevaluatedbyrepeatedsingleintegrationswithrespecttoρandφ

as27HowtoIntegrateinPolarCoordinateswhere(Ω)

istheregiondeterminedExampleEvaluatebytheinequalitiesxyOab(Ω):28FindingLimitsofIntegrationExampleFindthelimitsofintegrationforintegratingoverandoutsidethetheregion

(Ω)

thatliesinsidethecardioidcircleSolutionrightfigure.Step1:sketch.ShowintheStep3:theρ–limitsofintegrationStep2:theφ–limitsofintegrationThentheinteg