arc length - emaths弧长emaths

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1、ArcLengthAslongasyou’resatisfiedwithyourcalculusskills,arclengthisamatteroflogic.Tothe‘somewhere’onthispagethereisdiagramofagraphunderultra-magnificationanditshowsanapproximationtothelengthofasmallpieceofthecurve,.YoushouldnoticethatthisapproximationisasimplecaseofPythagorasan

2、dthat.Whatthismeansintermsofdifferentiationisthatifismadesmallerandsmallerandsmaller,untilit’ssosmallthatamoebaareusingitasflossthenwontbe‘approximately’,itwillbe.Atthispointthenicecurlydelta()signismadeintoa‘d’outofconventionsothat…Ifeverythingisdividedthroughbyandrearrangedy

3、ouwillfindyourselfwiththeequation:Noticeanythingfamiliar?It’sjustanordinaryolddifferentialisn’tit?Youcanintegrateitwithrespecttotoarriveat:Nowanexample:Findthearclengthofthecurvewithequationbetweenand.Isuggestwestartbyfinding,whichwecandobydifferentiatingtheequationtoseethat……

4、andthereforethat……intothisequationIcansubstitutein(thecurveequation)togetthatKnowingthesefacts,IcanbeginthemutationprocessofthearclengthintegraltoseethatI’llbefinding:Nobody,ifthey’rehonest,likestoseea4hangingbelowalikethat,soI’mjustgoingtonudgethelittlefellaoutoftheintegralin

5、abitofrearranging,asso…I’malsogoingtomakethesubstitution-whereandthenchangethelimitsoftheintegralsoshowthat:whichisequaltoNowtherestisasimpleintegrationjob;youshouldfindthat:…andthatyourgrandanswerisTakingthisalittlefurther–thelengthofanarcinparametricformcanbefoundbystartingf

6、romagain,butinsteadofdividingthroughbyyoushoulddividethroughbyandrearrangethingstogetthat:andthatI’llwarnyouthoughthatalotofparametricformquestions,oratleasttheonesinthebook,usuallyendupwithyouneedingtointegratesomethinglikewhereyouwillneedtousethesubstitution…thingsgetalittle

7、taxingafterthat,unlessyouremembertheintegralbyheart.Nowwecanuseournewarclengthknowledgeandapplyittosomethingslightlydifferentcalled‘areaofsurfacerevolution’.Itgoesalittlesomethinglikethis:Thisisjustlikevolumeofrevolutionbutinsteadlooksatthesurfacearea.Thesurfaceareaofacylinder

8、youshouldknowtobe,whereristheradiusofthecrosssectionalareaand