16.1Integration A-level考试纯数学P1课件.pptVIP

IntegrationChapter16

ContentsCalculateingareasTheareaalgorithm3Findingafunctionfromitsderivative12Somepropertiesofdefiniteintegrals4

01-1-2-345321xf’(x)f(x)x3x2xx3x2x?theprocessofgettingfromf’(x)tof(x)iscalledintegration,andthegeneralexpressionforf(x)iscalledtheindefiniteintegraloff’(x)

Theindefiniteintegralofafunctionmadeupofthesumofmultiplesofxn,wheren≠-1,isthecorrespondingsumofmultiplesof,togetherwithanaddedarbitraryconstant.

ExampleThegraphofy=f(x)passesthrough(2,3),andf’(x)=6x2–5x.Finditsequation.

ExampleAgardenerisdiggingaplotofland.Ashegetstiredheworksmoreslowly;aftertminutesheisdiggingatarateofsquaremetresperminute.Howlongwillittakehimtodiganareaof40squaremetres?

y14x=

TheareaalgorithmA=I(x)+kaby=f(x)x0=I(a)+kk=-I(a)A=I(x)–I(a)Tofindtheareaunderthegraphy=f(x)fromx=atox=b:Step1Findthe‘simplest’integraloff(x);callitI(x).Step2WorkoutI(a)andI(b).Step3TheareaI(b)–I(a).f(x)0axb

Findtheareaundery=x–2fromx=2tox=5.ExampleThe‘areaundery=f(x)fromx=atox=b’isdonatedbyThisiscalledadefiniteintegral.Thenumbersaandbareoftencalledthelimits,orthebounds,ofintegration.Thefunctionf(x)iscalledtheintegrand.

ThereisalsoanabbreviationforI(b)–I(a):itiswrittenExampleFindtheareaunderfromx=1tox=4

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