C3.6 Differentiation and Integration 2.ppt

theseiconsindicatethatteacher snotesorusefulwebaddressesareavailableinthenotespage thisiconindicatestheslidecontainsactivitiescreatedinflash theseactivitiesarenoteditable formoredetailedinstructions seethegettingstartedpresentation boardworksltd2006 1of40 a levelmaths core3forocr c3 6differentiationandintegration2 contents boardworksltd2006 2of40 integralsofstandardfunctionsreversingthechainruleintegrationbysubstitutionvolumesofrevolutionexamination stylequestion integralsofstandardfunctions reviewofintegration sofar wehaveonlylookedatfunctionsthatcanbeintegratedusing forexample integratewithrespecttox theintegralof adding1tothepowerandthendividingwouldleadtothemeaninglessexpression theonlyfunctionoftheformxnthatcannotbeintegratedbythismethodisx 1 thisdoesnotmeanthatcannotbeintegrated therefore theintegralof wecanonlyfindthelogofapositivenumberandsothisisonlytrueforx 0 wecangetaroundthisbytakingxtobenegative however doesexistforx 0 butnotx 0 sohowdoweintegrateitforallpossiblevaluesofx ifx0so wecancombinetheintegralsofforbothx 0andx 0byusingthemodulussigntogive theintegralof find thisisjusttheintegralofmultipliedbyaconstant find definiteintegralsinvolving itisparticularlyimportanttorememberthemodulussignwhenevaluatingdefiniteintegralsoffunctionsinvolving findtheareaunderthecurvey betweenx 3 x 1andthex axis writingyouranswerintheformlna 1 3 theareaisgivenby rememberthatln1 0 unitssquared definiteintegralsinvolving weshouldnotethatdefiniteintegralsoftheformcanonly beevaluatedifx 0doesnotlieintheinterval a b integralsofstandardfunctions byreversingtheprocessofdifferentiationwecanderivetheintegralsofsomestandardfunctions theseintegralsshouldbememorized integralsofstandardfunctions also ifanyfunctionismultipliedbyaconstantkthenitsintegralwillalsobemultipliedbytheconstantk find inpracticemostofthesestepscanbeleftout contents boardworksltd2006 11of40 integralsofstandardfunctionsreversingthechainruleintegrationbysubstitutionvolumesofrevolutionexamination stylequestion reversingthechainrule reversingthechainrule averyhelpfultechniqueistorecognizethatafunctionthatwearetryingtointegrateisofaformgivenbythedifferentiationofacompositefunction thisissometimescalledintegrationbyrecognition let bythechainrule so itfollowsthatforn 1 reversingthechainrule supposewewanttointegrate 2x 7 5withrespecttox iftheintegralismultipliedbyaconstantk considerthederivativeofy 2x 7 6 usingthechainrule 12 2x 7 5 so don ttrytolearnthisformula justtrytorecognizethatthefunctionyouareintegratingisoftheformk f x nf x andcompareittothederivativeof f x n 1 reversingthechainrule ingeneral youcanintegrateanylinearfunctionraisedtoapowerusingtheformula withpractice integralsofthistypecanbewrittendowndirectly forexample reversingthechainrule integratey x 3x2 4 3withrespecttox noticethatthederivativeof3x2 4is6x usingthechainrule 24x 3x2 4 3 so let slookatsomemoreintegralsoffunctionsoftheformk f x nf x nowconsiderthederivativeofy 3x2 4 4 reversingthechainrule nowconsiderthederivativeofy 2x3 9 3 so find noticethatthederivativeof2x3 9is6x2 reversingthechainrule so find startbywritingas nowconsiderthederivativeofy x2isthederivativeof x3 1 plus1is usingthechainrule reversingthechainruleforexponentialfunctions whenweappliedthechainruletofunctionsoftheformef x weobtainedthefollowinggeneralization wecanreversethistointegratefunctionsoftheformkf x ef x forexample anumericaladjustmentisusuallynecessary reversingthechainruleforexponentialfunctions ingeneral find reversingthechainruleforexponentialfunctions withpractice thismethodcanbeextendedtocaseswheretheexponentisnotlinear forexample find noticethatthederivativeof2x2is4xandsothefunctionweareintegratingisoftheformkf x ef x contents boardworksltd2006 21of40 integralsofstandardfunctionsreversingthechainruleintegrationbysubstitutionvolumesofrevolutionexamination stylequestion integrationbysubstitution integrationbysubstitution withpractice thetechniqueofintegrationbyrecognitioncansavealotoftime however whenitistoodifficulttouseintegrationbyrecognitionwecanuseamoreformalmethodofreversingthechainrulecalledintegrationbysubstitution toseehowthismethodworksconsidertheintegral letu 5x 2sothat theproblemnowisthatwecan tintegrateafunctioninuwithrespecttox wethereforeneedtowritedxintermsofdu integrationbysubstitution nowchangethevariablebacktox whenweusedthechainrulefordifferentiationwesawthatwecantreatinformallyasafraction so soifu 5x 2anddx integrationbysubstitution useasuitablesubstitutiontofind letu 2x2 5 substitutinguanddxintotheoriginalproblemgives noticethatthex scancelout integrationbysubstitution thisintegralcouldalsohavebeenfounddirectlybyrecognition nowweneedtochangethevariablebacktox however therearefunctionsthatcanbeintegratedbyuseofasuitablesubstitutionbutnotbyrecognition forinstance usethesubstitutionu 1 2xtofind ifu 1 2xthen integrationbysubstitution substitutingtheseintotheoriginalproblemgives wealsohavetosubstitutethexsothatthewholeintegrandisintermsofu alsoifu 1 2xthen integrationbysubstitution changingthevariablebacktoxgives definiteintegrationbysubstitution whenadefiniteintegralisfoundbysubstitutionitiseasiesttorewritethelimitsofintegrationintermsofthesubstitutedvariable ifu then usingthechainrulefordifferentiation definiteintegrationbysubstitution rewritethelimitsintermsofu nowweneedtofindxintermsofu u2 8 x x 8 u2 whenx 3 whenx 1 theareaisgivenby rewritethisintermsofu ifu then definiteintegrationbysubstitution therefore therequiredareaisunitssquared contents boardworksltd2006 31of40 integralsofstandardfunctionsreversingthechainruleintegrationbysubstitutionvolumesofrevolutionexamination stylequestion volumesofrevolution volumesofrevolution considertheareaboundedbythecurvey f x thex axisandx aandx b ifthisareaisrotated360 aboutthex axisathree dimensionalshapecalledasolidofrevolutionisformed thevolumeofthissolidiscalleditsvolumeofrevolution volumesofrevolution wecancalculatethevolumeofrevolutionbydividingthevolumeofrevolutionintothinslicesofwidth x volumesofrevolution thetotalvolumeofthesolidisgivenbythesumofthevolumeoftheslices thesmaller xis thecloserthisapproximateareaistotheactualarea wecanfindtheactualareabyconsideringthelimitofthissumas xtendsto0 thislimitisrepresentedbythefollowingintegral volumesofrevolution soingeneral thevolumeofrevolutionvofthesolidgeneratedbyrotatingthecurvey f x betweenx aandx baboutthex axisis similarly thevolumeofrevolutionvofthesolidgeneratedbyrotatingthecurvex f y betweeny aandy