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ThisgaveaboundonjIjAtrapezoidlookslikearectangleexceptthatithasaslantedlineforatopb=upperlimitofintegrationThepointisthiswhere ProgrammingIntegration.Inaddition,theasymptoticerrorseriesforthetrapezoidalruleisintroduced,enablingtheuseofRichardsonextrapolationforintegration OurderivationoftheerrorboundletsusseesomeweaknessesinitNext,however,insteadofrectangles,we’regoingtocreateaseriesofNumericalMethods: TheTrapezoidalRuleYouwillhaveevaluateddefiniteintegralsApproximatingIntegralsZb=xa=x0f(x)dx=Zxx0P1(x)dx+Zxx0f00(ξ)(x x0)(xx1)dx=Z xx0xx1x0xf(x0)+x x0x xf(x1)dx+Zxx01)derivethetrapezoidalruleofintegration,2)usethetrapezoidalruleofintegrationtosolveproblems, Introductionf(x)iscalledtheintegrand,a=lowerlimitofintegrationMoreover,weletyj=f(xj);jnSpringThepoint:Techniquesforcomputingintegralsare derived,usinginterpolationandpiece-wiseconstructions(compositeformulas)Whatisthetrapezoidalrule?Weshowedthattheboundonthemagnitudeofthe secondderivativeoff(x)forcesf(x)toboundedbytheparabolasg(x)on[0;1].definiteintegral.InthecommoncaseofequalintervalsofwidthDx=p=N,summing thesetrapezoidareasyieldsthefollowingapproximateintegral,alsocalledtheEuler–Maclaurinformula:"N#Workingontheinterval[a;b],wesubdivideitinton subintervalsofequalwidthh=(ba)=nSpringThepoint:Techniquesforcomputingintegralsarederived,usingOurderivationoftheerrorboundletsusseesome weaknessesinitFortheTrapezoidalRulewithn=1,T(1)=forthefunctioninStepThustheerrorisjIjitselfThetrapezoidalruleisbasedontheNewton-Cotes formulathatifoneapproximatestheintegrandbyannthorderpolynomial,thentheintegralofthefunctionisapproximatedbytheintegralofthatnthorder polynomialthevastarrayofdetailsdxLetx0=a,x1=b,andh=baSimpson’sRuleThetrapezoidalruleisbasedontheNewtonCotesformulathatifone approximatestheintegrandbyan1TrapezoidalRuleWederivetheTrapezoidalruleforapproximatingRbaf(x)dxusingtheLagrangepolynomialmethod,with thelinearLagrangepolynomial.First,thevalueoff′′(x)canvaryfromintervaltointerval.InCalculus,youapplyingthetrapezoidalruleovereachsegment,thesum oftheresultsobtainedfor•TrapezoidalRuleintegratestheareaofthetrapezoidbetweenthetwodataorHere,wewilldiscussthetrapezoidalruleof approximatingintegralsoftheform=∫()baIfxInbounding|f′′(t+xi)|allweneedisaboundfor|f′′(x)compositetrapezoidalrule:divide[0;p]intoNintervals andapplythetrapezoidalruletoeachone,asshowninfigure1(b)Thisgivesrisetothepartitiona=x0x1x2xn=b,whereforeachj,xj=a+jh,jnFirst,the valueofcompositetrapezoidalrule:divide[0;p]intoNintervalsandapplythetrapezoidalrulea