《高数双语》课件section 5-1

Section5.1ConceptsandPropertiesofDefiniteIntegrals[定积分]12ExampleofDefiniteIntegralProblemsComputingtheareasurroundedbythecurveExample(Areaofatrapezoidwithacurvedtop),

andthehorizontallinetheverticallines3ExampleofDefiniteIntegralProblemsComputingtheareasurroundedbythecurveExample(Areaofatrapezoidwithacurvedtop),

andthehorizontallinetheverticallines4ExampleofDefiniteIntegralProblemsComputingtheareasurroundedbythecurveExample(Areaofatrapezoidwithacurvedtop),

andthehorizontallinetheverticallines5ExampleofDefiniteIntegralProblemsArbitrarilyinsertn-1pointsofdivision,Betweenaandb,suchthatSolution

(1)“partition”,Ifwedenote

and,andComputingtheareasurroundedbythecurveExample(Areaofatrapezoidwithacurvedtop),

andthehorizontallinetheverticallines6ExampleofDefiniteIntegralProblemsSolution:becomesverysmall.

and

asitsheight.(2)“homogenization”

isverysmall,WhilethevarianceoffunctiononapproximatetheareaoftrapezoidbychoosinganypointSo,wecanusetherectangletoComputingtheareasurroundedbythecurveExample(Areaofatrapezoidwithacurvedtop),

andthehorizontallinetheverticallines7ExampleofDefiniteIntegralProblemsDoingthesamethingforeachsubinterval,andthencombiningalltheapproximatevalues,wehaveSolution(3)“summation”ComputingtheareasurroundedbythecurveExample(Areaofatrapezoidwithacurvedtop),

andthehorizontallinetheverticallines8ExampleofDefiniteIntegralProblemsIfwerefinethepartitionsof[a,b],thesum,

isacloserapproximationtothetotalarea.Therefore,where,Solution:(4)“precision”ComputingtheareasurroundedbythecurveExample(Areaofatrapezoidwithacurvedtop),

andthehorizontallinetheverticallinesThedefinitionofdefiniteintegral910ThedefinitionofdefiniteintegralRiemannsum11Thedefinitionofdefiniteintegral12Thedefinitionofdefiniteintegral13ThedefinitionofdefiniteintegralNote

Wealsodefine14ConditionsforexistenceforthedefiniteintegralTheorem(necessaryconditionforintegrability)

If

isintegrableovertheinterval,then

mustbe.boundedonInotherwords,if

isunboundedonthenitisnotintegrable.15ConditionsforexistenceforthedefiniteintegralTheorem(sufficientconditionforintegrability)afinitenumberoffirsttypediscontinuouspoints,then

mustbeintegrableovertheinterval.

iscontinuousontheintervalIf,orhasonlyabc16ConditionsforexistenceforthedefiniteintegralSolution:Wepartition

into

nequalsubintervals,thenthepointsoftheand

iscontinuousontheintervalSincethefunctionitisintegrable.theRiemannsum.partitionareThenwecanselectaspecialpartitiontoconstructExample

Findbythedefinitionofthedefiniteintegral.Conditionsforexistenceforthedefiniteintegral17Solution(continued)thentheRiemannsumisTherefore,Let..Example

Findbythedefinitionofthedefiniteintegral.18Conditionsforexistenceforthedefiniteintegral

Wechoose

betweenIteasytoseethatIfwechoosethenIfwelet,wehave.Solution

andthesepointsdividetheinterval

into

subintervals...19ConditionsforexistenceforthedefiniteintegralSolution(continued)so,wehave

andthismeansthatFinish.Since.20GeometricinterpretationofthedefiniteintegralIf,then21PropertiesofDefiniteIntegralsBythedefinitionofdefiniteintegral,wecallthiskindofdefiniteintegralasRiemannintegral

andwedenotethesetofallfunctionswhichareintegrableovertheinterval

by.Thispropertycanbeeasilyprovedbythedefinitionofthedefiniteintegral.22PropertiesofDefiniteIntegralsWeselectapartitionsuchthat

isalwayskeptasapointofthepartition.Let,weobtain.Proof(1)Assumethat.Thus23PropertiesofDefiniteIntegralssay,,thenwehaveSothatSince,thuswehavetheconclusion.Proof(continued)(2)Supposethat

isoutsidetheinterval,..24GeometricinterpretationofthedefiniteintegralPropertiesofDefiniteIntegrals25Example

Findbyitsgeometricmeaning.yOx126PropertiesofDefiniteIntegrals27PropertiesofDefiniteIntegralsSolution：ExampleFindtheboundforPropertiesofDefiniteIntegrals28Proof:

ofthefunctionontheclosedinterval,andwehaveSincewehaveknownthatwehave.,theremusthavethemaximum

andminimumSince.PropertiesofDefiniteIntegrals29MeanvalueProof(continued)minimumandmaximumof.Bytheintermediatevaluetheorem,we

suchthatandthisimpliestheconclusion.Andthismeansthat

isavaluebetweenthehavethatthereexistsatleastone