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1、Prascius11StevenPrasciusMrs.TallmanAPCalculus17March2014RiemannSums,TrapezoidRule,andSimpsonsRuleWhendeterminingtheareaunderthecurveofafunction,thedefiniteintegralisalmostalwaysused.Althoughthedefiniteintegralisbyfarthemostaccuratemethodfordeterminingtheareaundera
2、curve,therearemanyothermethods.TheseothermethodsareRiemannsums,thetrapezoidrule,andSimpson’srule.OnemethodforfindingtheareaunderthecurveisRiemannsums.ARiemannsumapproximatestheareaunderthecurvebyseparatingtheareaunderthecurveintoseparaterectanglesandaddingtheareas
3、oftherectanglestogether.TheformalnotationofaRiemannsumisRn=i=1nfx*∆xwhere∆x=(b-a)/n.Inthisnotationnisthenumberofintervals,ornumberofrectanglethatwillbeused,aandbarethestartandendpointsfortheintegral,andf(x)istheheightoftherectangle,ory-value.Thisnotationsimplymea
4、nsthattheareasoftherectangles,foundbymultiplyingheight(f(x))bywidth(∆x),areaddedtogethertogetanestimatefortheareaunderthecurve.TherearefivedifferenttypesofRiemannsumswhichareleft,right,midpoint,upper,andlowerRiemannsums.AllofthesetypesofRiemannsumsarefoundusingrec
5、tangleswiththeonlydifferencebeingtheheightoftherectanglestheyuse.AllofthetypesofRiemannsumsusethesameintervalandsamewidthsoftherectanglesbutusedifferentheightsfoundatdifferentpartsoftheintegrals.LeftRiemannsumsuseheights,orf(x)values,thatarefoundontheleftmostporti
6、onofeachintervalwhilerightRiemannsumsuseheightsthatarefoundontherightmostportionofeachinterval.MidpointRiemannsumsusePrascius11theheightvaluefounddirectlyinthecenteroftheinterval.UpperRiemannsumsusetheheightsthatcorrespondtothehighestf(x)valueintheintegralandlower
7、Riemannsumsuseheightsthatcorrespondtothelowestf(x)valueintheintegral.ThedifferencesbetweenthetypesofRiemannsumscanbeseeninfigure1below.MidpointRiemannSumRightRiemannSumLeftRiemannSumLowerRiemannSumUpperRiemannSumFigure1.TypesofRiemannSumsAsseeninthefigureabove,dif
8、ferenttypesofRiemannsumsoverorunderestimatetheareaunderthecurvedependingontheshapeofthefunction.BecauseofthisitisimportanttochooseatypeofRiemannsumthatw