fundamental theorem of calculus：微积分基本定理

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1、5.4FundamentalTheoremofCalculusGregKelly,HanfordHighSchool,Richland,WashingtonPhotobyVickieKelly,1998MorroRock,CaliforniaIfyouwerebeingsenttoadesertislandandcouldtakeonlyoneequationwithyou,mightwellbeyourchoice.Hereismyfavoritecalculustextbookquoteofalltime,fromCALCULUSbyRossL.Finneya

2、ndGeorgeB.Thomas,Jr.,©1990.TheFundamentalTheoremofCalculus,Part1Iffiscontinuouson,thenthefunctionhasaderivativeateverypointin,andFirstFundamentalTheorem:1.Derivativeofanintegral.2.Derivativematchesupperlimitofintegration.FirstFundamentalTheorem:1.Derivativeofanintegral.1.Derivativeofa

3、nintegral.2.Derivativematchesupperlimitofintegration.3.Lowerlimitofintegrationisaconstant.FirstFundamentalTheorem:1.Derivativeofanintegral.2.Derivativematchesupperlimitofintegration.3.Lowerlimitofintegrationisaconstant.Newvariable.FirstFundamentalTheorem:1.Derivativeofanintegral.2.Der

4、ivativematchesupperlimitofintegration.3.Lowerlimitofintegrationisaconstant.Thelongway:FirstFundamentalTheorem:1.Derivativeofanintegral.2.Derivativematchesupperlimitofintegration.3.Lowerlimitofintegrationisaconstant.Theupperlimitofintegrationdoesnotmatchthederivative,butwecouldusethech

5、ainrule.Thelowerlimitofintegrationisnotaconstant,buttheupperlimitis.Wecanchangethesignoftheintegralandreversethelimits.Neitherlimitofintegrationisaconstant.Itdoesnotmatterwhatconstantweuse!(Limitsarereversed.)(Chainruleisused.)Wesplittheintegralintotwoparts.TheFundamentalTheoremofCalc

6、ulus,Part2Iffiscontinuousateverypointof,andifFisanyantiderivativeoffon,then(AlsocalledtheIntegralEvaluationTheorem)Wealreadyknowthis!Toevaluateanintegral,taketheanti-derivativesandsubtract.p