《高數(shù)雙語(yǔ)》課件section 5-5

Section5.5ApplicationsofDefiniteIntegralsMethodofelementsforsettingupintegralrepresentationsWhatkindofquantitiescanbecalculatedbydefiniteintegrals?Howcanwesetuptheintegralrepresentations?2QuestionsManyquantitieswewanttoknowinscienceandtechnologycanbecalculatedbydefiniteintegrals.Methodofelementsforsettingupintegralrepresentations3Wehadseenthattheareaoftrapezoidwithcurvedtop,themassofastickandthedisplacementofamovementalongastraightlinecanallbeexpressedbydefiniteintegrals.Theybothhavethefollowingproperties.1)Theyarealldistributednon-uniformlybutcontinuouslyonaninterval[a,b]2)Thesequantitiesarealladditive,thatis,thetotalquantityontheinterval[a,b]equalsthesumofallthoselocalquantitiesdistributedonthesubintervalsof[a,b].Ingeneral,aquantitywiththesetwocharacteristicsmaybecalculatedbyadefiniteintegral.Methodofelementsforsettingupintegralrepresentations4Inordertocalculatethewholearea,wewillfirstfindtheapproximatevalueofeverylocalarea.Todothis,weneedtwosteps:(1)“partition”Divide[a,b]intomanyverysmallsubintervals;(2)“homogenization”Regardthelocalareadistributedoneachsmallinterval[x,x+Δx]asarectanglewithheightf(x).Usingmultiplicationfortheuniformdistribution,wehaveMethodofelementsforsettingupintegralrepresentations5Afterwefoundtheapproximationvalueofeverylocalarea,wecaneasilyobtaintheprecisevalueofthewholeareaAbytheothertwosteps:(3)“summation”(4)“precision”Becausethefunction

f(x)iscontinuouson[a,b],thelimitofthesumisjustthedefiniteintegral.Methodofelementsforsettingupintegralrepresentations6Sothelocalarea

isjusttheincrementofthisfunction.SinceBythegeometricmeaningofthedefiniteintegralwithvaryingupperlimit,weknownthattheareaofthedefiniteintegralwiththecurve

y=f(x)andontheInterval[a,x]isexpressedby,namelytheapproximatevalueofthelocalarea,isactuallythedifferentialofthefunction(1).Methodofelementsforsettingupintegralrepresentations7Ingeneral,theproceduresmaybesimplifiedintothefollowingtwosteps:(1)FindtheelementFindtheapproximatevalueofthelocalrequiredquantity,dQ,onthesubinterval(2)SetuptheintegrationForthedifferential,writedownthecorrespondingdefiniteintegralovertheinterval,weobtainInfinitesimalelementoftheintegralorelementoftheintegralTheproceduresummarizedbytheabovetwostepsiscalledtheelementmethodofintegration.TheAreaofaPlaneRegion8ExampleFindtheareaAoftheregionenclosedbytheparabolasandSolutionFromthesystemwecaneasilyobtaintheabscissaeoftheintersectionsofthetwoparabolas,ItiseasytoseethattheareaAisdistributedcontinuouslybutnon-uniformontheinterval[-2,2],andisadditive.TheAreaofaPlaneRegion9Solution(continued)Bythemethodofelement,wecalculateA

bytwosteps:(1)FindtheelementPartitiontheinterval[-2,2]andconsiderthesubinterval[x,x+dx].Onthissubinterval,theareadistributednon-uniformlymayberegardedapproximatelyasuniform,thatisarectanglewithheightExampleFindtheareaAoftheregionenclosedbytheparabolasandTheAreaofaPlaneRegion10Solution(continued)Thus,weobtaintheelementofarea:ExampleFindtheareaAoftheregionenclosedbytheparabolasand(1)Findtheelement(2)SetuptheintegralTheAreaofaPlaneRegion11Solution(continued)overtheinterval[-2,2].ThewholeareaAisjusttheintegraloftheelementSoFinish.ExampleFindtheareaAoftheregionenclosedbytheparabolasandTheAreaofaPlaneRegion12ExampleFindtheareaAoftheregionenclosedbytheparabolaandthestraightlinesSolutionThisareaAmayberegardedasaquantitywhichisdistributednon-uniformlyontheinterval[0,1]onthey-axis.Tofindtheelementofthearea,partitiontheinterval[0,1],andconsiderthesubinterval,regardingastheareaofarectanglewithwidthTheAreaofaPlaneRegion13Solution(continued)Hence,thetotalareaisThentheelementoftheareaisFinish.ExampleFindtheareaAoftheregionenclosedbytheparabolaandthestraightlinesTheAreaofaPlaneRegion14ExampleFindtheareaAoftheregionenclosedbythecardioidSolutionBythesymmetryofthegraphofthecardioid,itisenoughtocalculatethearealocatedintheupperhalf-plane.Sincetheequationofthecardioidisexpressedinpolarcoordinates,theareamayberegardedasadistributionontheinterval[0,π]andthedistributionisnon-uniform.TheAreaofaPlaneRegion15Solution(continued)Tocalculatethearea,wepartitiontheinterval[0,π]andconsiderthesubintervalOnthisinterval,weregardasAconstant.ThustheelementofareaisExampleFindtheareaAoftheregionenclosedbythecardioidTheAreaofaPlaneRegion16Solution(continued)Thus,thetotalareaisFinish.ExampleFindtheareaAoftheregionenclosedbythecardioidTheAreaofaPlaneRegion17yxOExampleFindtheareaAoftheregionenclosedbythecurveandthex-axis.Solution18TheArcLengthofaPlaneCurveTostudymotionalongaspacecurve,weneedtohavea

measurablelengthalongthecurve.xyOM0Mba19TheArcLengthofaPlaneCurvexyOMi-1Mi(1)Findtheelementofthearclength20TheArcLengthofaPlaneCurve(2)SetuptheintegrationxyOMi-1Mi21TheArcLengthofaPlaneCurveArcLengthforasmoothcurveinthecaseoftherectangularcoordinates1)IftheequationoftheplanecurveΓis

thenthelengthofthearcΓis

2)IftheequationoftheplanecurveΓis

thenthelengthofthearcΓis

TheArcLengthofaPlaneCurve22ExampleFindthearclengthoftheplanecurve23TheArcLengthofaPlaneCurveExample

Findthelengthofanarcoftheplanecurve24TheArcLengthofaPlaneCurveArcLengthforaSmoothCurveinthecaseoftheparametricform

IftheequationoftheplanecurveΓis

thenthelengthofthearcΓis

whereφ(t)andψ(t)arecontinuouslyderivable,and(φ`(t),ψ`(t))≠0,25ExampleFindthelengthofanarcofthecycloidTheArcLengthofaPlaneCurve26TheArcLengthofaPlaneCurveArcLengthforaSmoothCurveinthecaseofthepolarcoordinatesIfΓisexpressedinpolarcoordinates,thenthelengthofthearcΓis

27TheArcLengthofaPlaneCurveExampleFindthelengthofanarcoftheplanecurveTheVolumeofaSolid28ExampleConsiderasolidliketheoneshowninthefollowingfigure.Ateachthecrosssectionofthesolidisaregionwhoseareaisaknowncontinuousfunction.Expressthevolumeofthissolidbyanintegration.SolutionWepartition[a,b]sothatthesolidcanbecutintomanyslicesbyperpendicularplanesthroughthePointsofthepartition.Considertheslicecorrespondingtothesubinterval.Wehavetheelementofvolume(1)Findtheelementofthevolume.TheVolumeofaSolid29Solution(continued)Bythemethodofelementsweobtainthevolumeofthegivensolidasfollows(2)Setuptheintegration.ExampleConsiderasolidliketheoneshowninthefollowingfigure.Ateachthecrosssectionofthesolidisaregionwhoseareaisaknowncontinuousfunction.Expressthevolumeofthissolidbyanintegration.TheVolumeofaSolid30HowtoFindVolumesbytheMethodofSlicingStep1.Sketchthesolidandatypicalcrosssection.Step2.FindaformulaforA(x).Step3.Findthelimitsofintegration.Step4.IntegrateA(x)tofindthevolume.TheVolumeofaSolid31ExampleAcurvedwedgeiscutfromacylinderofradius3bytwoplanes.Oneplaneisperpendiculartotheaxisofthecylinder.Thesecondplanecrossesthefirstplaneata45oangleatthecenterofthecylinder.Findthevolumeofthewedge.SolutionWedrawthewedgeandsketchatypicalcrosssectionperpendiculartothex-axis.Thecrosssectionatxisarectangleofarea(1)Asketch.(2)TheformulaforA(x).TheVolumeofaSolid32Solution(continued)(4)Integratetofindthevolume.Therectanglesrunfromto(3)Thelimitsofintegration.Finish.ExampleAcurvedwedgeiscutfromacylinderofradius3bytwoplanes.Oneplaneisperpendiculartotheaxisofthecylinder.Thesecondplanecrossesthefirstplaneata45oangleatthecenterofthecylinder.Findthevolumeofthewedge.TheVolumeofaSolid33Themostcommonapplicationofthemethodofslicingistosolidsofrevolution[旋轉(zhuǎn)體].SolidsofrevolutionaresolidswhoseshapescanbegeneratedbyrevolvingplaneRegionsaboutaxes.TheonlythingthatchangeswhenthecrosssectionsarecircularistheformulafortheareaA(x).TheVolumeofaSolid34ThetypicalcrosssectionofthesolidperpendiculartotheaxisofrevolutionisadiskofradiusR(x)andareaForthisreason,themethodisoftencalledthediskmethod.TheVolumeofaSolid35ExampleTheregionbetweenthecurve,andthex-axisisrevolvedaboutthex-axistogenerateasolid.Finditsvolume.TheVolumeofaSolid36SolutionWedrawfiguresshowingtheregion,atypicalradius,andthegeneratedsolid.ThevolumeisFinish.ExampleTheregionbetweenthecurve,andthex-axisisrevolvedaboutthex-axistogenerateasolid.Finditsvolume.TheVolumeofaSolid37ExampleFindthevolumeofthesolidgeneratedbyrevolvingaregionboundedbyandthelines,abouttheline.TheVolumeofaSolid38SolutionWedrawfiguresshowingtheregion,atypicalradius,andthegeneratedsolid.ThevolumeisFinish.ExampleFindthevolumeofthesolidgeneratedbyrevolvingaregionboundedbyandthelines,abouttheline.TheVolumeofaSolid39HowtoFindVolumesforCircularCrossSections(DiskMethod)Step1.DrawtheregionandidentifytheradiusfunctionR(x).Step2.SquareR(x)andmultiplybyπ.Step3.Integratetofindthevolume.TheVolumeofaSolid40ExampleFindthevolumeofthesolidgeneratedbyrevolvingaregionbetweenthey-axisandthecurve,,aboutthey-axis.TheVolumeofaSolid41SolutionWedrawfiguresshowingtheregion,atypicalradius,andthegeneratedsolid.ThevolumeisFinish.ExampleFindthevolumeofthesolidgeneratedbyrevolvingaregionbetweenthey-axisandthecurve,,aboutthey-axis.TheVolumeofaSolid42Iftheregionwerevolvetogenerateasoliddoesnotborderonorcrosstheaxisofrevolution,thesolidhasaholeinit.Thecrosssectionsperpendiculartotheaxisofrevolutionarewashersinsteadofdisks.ThedimensionsofatypicalwasherareOuterradius:Innerradius:Thewasher’sareaisTheVolumeofaSolid43HowtoFindVolumesforWasherCrossSectionsStep1.Drawtheregionandsketchalinesegmentacrossitperpendiculartotheaxisofrevolution.Whentheregionisrevolved,thissegmentwillgenerateatypicalwashercrosssectionofthegeneratesolid.Step2.Findthelimitsofintegration.

Step3.Findtheouterandinnerradiiofthewashersweptoutbythelinesegment.Step4.Integratetofindthevolume.TheVolumeofaSolid44ExampleTheregionboundedbythecurveandthelineisrevolvedaboutthex-axistogenerateasolid.Findthevolumeofthesolid.Solution

Step1.Drawtheregionandsketchalinesegmentacrossitperpendiculartotheaxisofrevolution.TheVolumeofaSolid45Solution(continued)

Step2.Findthelimitsofintegrationbyfindingthex-coordinatesoftheintersectionpointsofthecurveandthelineinrightfigure.ExampleTheregionboundedbythecurveandthelineisrevolvedaboutthex-axistogenerateasolid.Findthevolumeofthesolid.TheVolumeofaSolid46Solution(continued)

Outerradius:Innerradius:Step3.Findtheouterandinnerradiiofthewasherthatwouldbesweptoutbythelinesegmentifitwererevolvedaboutthex-axisalongwiththeregion.ExampleTheregionboundedbythecurveandthelineisrevolvedaboutthex-axistogenerateasolid.Findthevolumeofthesolid.TheVolumeofaSolid47Solution(continued)Step4.Evaluatethevolumeintegral.Finish.ExampleTheregionboundedbythecurveandthelineisrevolvedaboutthex-axistogenerateasolid.Findthevolumeofthesolid.Theapplicationsofthedefiniteintegralinphysics48PumpingLiquidsfromContainersHowmuchworkdoesittaketopumpallorpartoftheliquidfromacontainer?Tofindout,weimagineliftingtheliquidoutonethinhorizontalslabatatimeandapplyingtheequationW=Fdtoeachslab,whereFistheforceanddisthedistanceoftheobjectalongwiththedirectionofF.Wethenevaluatetheintegralthisleadtoastheslabsbecomethinnerandmorenumerous.Theintegralwegeteachtimedependsontheweightoftheliquidandthedimensionsofthecontainer,butthewaywefindtheintegralisalwaysthesame.Theapplicationsofthedefiniteintegralinphysics49ExampleHowmuchworkdoesittaketopumpthewaterfromafulluprightcircularcylindricaltankofradius5mandheight10mtoalevelof4mabovethetopofthetank?SolutionWedrawthetankasrightfigure,addcoordinateaxes,andi