mit 单变量微积分笔记 英文版(下)

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1、MITOpenCourseWarehttp://ocw.mit.edu18.01SingleVariableCalculusFall2006ForinformationaboutcitingthesematerialsorourTermsofUse,visit:http://ocw.mit.edu/terms.Lecture1818.01Fall2006Lecture18:DefiniteIntegralsIntegralsareusedtocalculatecumulativetotals,averages,a

2、reas.Areaunderacurve:(SeeFigure1.)1.Divideregionintorectangles2.Addupareaofrectangles3.Takelimitasrectanglesbecomethinabab(i)(ii)Figure1:(i)Areaunderacurve;(ii)sumofareasunderrectanglesExample1.f(x)=x2,a=0,barbitrary1.DivideintonintervalsLengthb/n=baseofrect

3、angle2.Heights:��2stbb•1:x=,height=nn��2nd2b2b•2:x=,height=nnSumofareasofrectangles:����2����2����2����23bbb2bb3bbnbb2222+++···+=(1+2+3+···+n)nnnnnnnnn31Lecture1818.01Fall2006a=0bFigure2:Areaunderf(x)=x2above[0,b].Wewillnowestimatethesumusingsome3-dimensiona

4、lgeometry.ConsiderthestaircasepyramidaspicturedinFigure3.n=4nFigure3:Staircasepyramid:left(topview)andright(sideview)1stlevel:n×nbottom,representsvolumen2.2ndlevel:(n−1)×(n−1),representsvolumne(n−1)2),etc.Hence,thetotalvolumeofthestaircasepyramidisn2+(n−1)2+

5、···+1.Next,thevolumeofthepyramidisgreaterthanthevolumeoftheinnerprism:222112131+2+···+n>(base)(height)=n·n=n333andlessthanthevolumeoftheouterprism:22212131+2+···+n<(n+1)(n+1)=(n+1)332Lecture1818.01Fall2006Inall,11n312+22+···+n21(n+1)3=3<<3n3n33n3Therefore,b3

6、1lim(12+22+32+···+n2)=b3,n→∞n33b3andtheareaunderx2from0tobis.3Example2.f(x)=x;areaunderxabove[0,b].ReasoningsimilartoExample1,buteasier,givesasumofareas:b21(1+2+3+···+n)→b2(asn→∞)n22ThisistheareaofthetriangleinFigure4.bbFigure4:Areaunderf(x)=xabove[0,b].Patt

7、ern:��db3=b2db3��db2=bdb2TheareaA(b)underf(x)shouldsatisfyA�(b)=f(b).3Lecture1818.01Fall2006GeneralPicturey=f(x)acibFigure5:OnerectanglefromaRiemannSumb−a•Divideintonequalpiecesoflength=Δx=n•Pickanyciintheinterval;usef(ci)astheheightoftherectangle•Sumofareas

8、:f(c1)Δx+f(c2)Δx+···+f(cn)Δx�nInsummationnotation:f(ci)Δx←calledaRiemannsum.i=1Definition:�n�blimf(ci)Δx=f(x)dx←calledadefiniteintegraln→∞i=1aThisdefiniteintegralrepresentstheareaunderthecurvey=f(x