Calculus II Review.pdf

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1、——5.1Antidifferentiation:TheIndefiniteIntegral——AfunctionF(x)issaidtobeanantiderivativeoff(x)ifF′(x)=f(x)Foreveryxinthedomainoff(x).Theprocessoffindingantiderivativesiscalledantidifferentiationorindefiniteintegration.RulesforIntegratingCommonFunctions:

2、∫kdx=kx+Ck:constantn+1nx∫xdx=+Cn≠-1n+11∫dx=lnx+Cx≠0xkx1kx∫edx=e+Ck≠0,constantkaxbax∫bedx=e+CaAlgebraicRulesforIndefiniteIntegration:∫kf(x)dx=k∫f(x)dx∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx∫[f(x)−g(x)]dx=∫f(x)dx−∫g(x)dxa∫f(x)dx=0aba∫f(x)dx=−∫f(x)dxabbcb∫f(x)dx=∫

3、f(x)dx+∫f(x)dxaacIntegrationistheinverseofdifferentiation:d∫[f(x)]dx=f(x)+CdxDifferentiationistheinverseofintegration:d[∫f(x)dx]=f(x)dx——5.2IndefiniteIntegrationbySubstitution——∫f(x)dx=∫g(u(x))u′(x)dxSupposeGisanantiderivativeofg,sothatG′=g.Then,accord

4、ingtothechainruled[G(u(x))]=G′(u(x))u′(x)dx=g(u(x))u′(x),sinceG′=gdxTherefore,byintegratingbothsidesofthisequationwithrespecttox,wefindthat∫f(x)dx=∫g(u(x))u′(x)dxd=∫([G(u(x))])dxdx=G(u(x))+C,since∫G′=GInotherwords,oncewehaveanantiderivativeforg(u),weal

5、sohaeoneforf(x).�UsingSubstitutiontoIntegrate∫f(x)dx:Step1.Chooseasubstitutionu=u(x)that“simplifies”theintegrandf(x).Step2.Expresstheentireintegralintermsofuanddu=u′(x)dx.Thismeansthatalltermsinvolvingxanddxmustbetransformedtotermsinvolvinguanddu.Step3

6、.Whenstep2iscomplete,thegivenintegralshouldhavetheform∫f(x)dx=∫g(u)duIfpossible,evaluatethistransformedintegralbyfindinganantiderivativeG(u)forg(u).Step4.Replaceubyu(x)inG(u)toobtainanantiderivativeG(u(x))forf(x),sothat∫f(x)dx=G(u(x))+C——5.3TheFundamen

7、talTheoremofCalculus——Ifthefunctionf(x)iscontinuousontheintervala≤x≤b,thenb∫f(x)dx=F(b)−F(a)awhereF(x)isanyantiderivativeoff(x)ona≤x≤b.——5.4ApplyingDefiniteIntegrationAreaBetweenCurvesandAverageValue——�TheAreaBetweenTwoCurves:Iff(x)andg(x)arecontinuous

8、withf(x)≥g(x)ontheintervala≤x≤b,thentheareaAbetweenthecurvesy=f(x)andy=g(x)overtheintervalisgivenbybA=∫[f(x)−g(x)]dxa�LorentzCurvesandGiniIndex:AreaalsoplaysanimportantroleinthestudyofLorentzcurves,adeviceusedbybotheconomistsandsociolog