The Trapezoidal Rule


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1 _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson s Rule. The Trpezoidl Rule Some elementr functions simpl do not hve ntiderivtives tht re elementr functions. For emple, there is no elementr function tht hs n of the following functions s its derivtive., cos, If ou need to evlute definite integrl involving function whose ntiderivtive cnnot e found, the Fundmentl Theorem of Clculus cnnot e pplied, nd ou must resort to n pproimtion technique. Two such techniques re descried in this section. One w to pproimte definite integrl is to use n trpezoids, s shown in Figure.. In the development of this method, ssume tht f is continuous nd positive on the intervl,. So, the definite integrl f d represents the re of the region ounded the grph of f nd the is, from to. First, prtition the intervl, into n suintervls, ech of width n, such tht < < <... < n. Then form trpezoid for ech suintervl (see Figure.). The re of the ith trpezoid is Are of i th trpezoid This implies tht the sum of the res of the n trpezoids is Are n f f... f n f n n n Letting n, ou cn tke the limit s n to otin lim n n f f... f n f n f f lim n n f i i f f lim lim n n n n f i i f d. f i f i n. f f f f... f n f n f f f... f n f n. The result is summrized in the following theorem. cos,, sin
2 _.qd // : PM Pge CHAPTER Integrtion THEOREM. The Trpezoidl Rule Let f e continuous on,. The Trpezoidl Rule for pproimting f d is given f d n f f f... f n f n. Moreover, s n, the righthnd side pproches f d. NOTE Oserve tht the coefficients in the Trpezoidl Rule hve the following pttern.... EXAMPLE Approimtion with the Trpezoidl Rule = sin Use the Trpezoidl Rule to pproimte sin d. Compre the results for n nd n, s shown in Figure.. Four suintervls = sin Trpezoidl pproimtions Figure. 5 7 Eight suintervls Solution When n,, nd ou otin When n,, nd ou otin sin d sin d sin sin sin sin sin sin sin sin sin 5 sin 7 sin sin sin sin.97. For this prticulr integrl, ou could hve found n ntiderivtive nd determined tht the ect re of the region is..9. sin sin TECHNOLOGY Most grphing utilities nd computer lger sstems hve uiltin progrms tht cn e used to pproimte the vlue of definite integrl. Tr using such progrm to pproimte the integrl in Emple. How close is our pproimtion? When ou use such progrm, ou need to e wre of its limittions. Often, ou re given no indiction of the degree of ccurc of the pproimtion. Other times, ou m e given n pproimtion tht is completel wrong. For instnce, tr using uiltin numericl integrtion progrm to evlute d. Your clcultor should give n error messge. Does ours?
3 _.qd // : PM Pge SECTION. Numericl Integrtion It is interesting to compre the Trpezoidl Rule with the Midpoint Rule given in Section. (Eercises ). For the Trpezoidl Rule, ou verge the function vlues t the endpoints of the suintervls, ut for the Midpoint Rule ou tke the function vlues of the suintervl midpoints. f d n i f d n f i i i f i f i Midpoint Rule Trpezoidl Rule NOTE There re two importnt points tht should e mde concerning the Trpezoidl Rule (or the Midpoint Rule). First, the pproimtion tends to ecome more ccurte s n increses. For instnce, in Emple, if n, the Trpezoidl Rule ields n pproimtion of.99. Second, lthough ou could hve used the Fundmentl Theorem to evlute the integrl in Emple, this theorem cnnot e used to evlute n integrl s simple s ecuse sin sin d hs no elementr ntiderivtive. Yet, the Trpezoidl Rule cn e pplied esil to this integrl. Simpson s Rule One w to view the trpezoidl pproimtion of definite integrl is to s tht on ech suintervl ou pproimte f firstdegree polnomil. In Simpson s Rule, nmed fter the English mthemticin Thoms Simpson (7 7), ou tke this procedure one step further nd pproimte f seconddegree polnomils. Before presenting Simpson s Rule, we list theorem for evluting integrls of polnomils of degree (or less). THEOREM.7 Integrl of p A B C If p A B C, then p d p p p. Proof p d A B C d B epnsion nd collection of terms, the epression inside the rckets ecomes A B C A B C A B C p p p d p p p. nd ou cn write A A B C B C A B C p
4 _.qd // : PM Pge CHAPTER Integrtion p (, ) (, ) f To develop Simpson s Rule for pproimting definite integrl, ou gin prtition the intervl, into n suintervls, ech of width n. This time, however, n is required to e even, nd the suintervls re grouped in pirs such tht < < < < <... < n < n < n.,, n, n (, ) Figure. n p d f d On ech (doule) suintervl i, i, ou cn pproimte f polnomil p of degree less thn or equl to. (See Eercise 55.) For emple, on the suintervl,, choose the polnomil of lest degree pssing through the points,,,, nd,, s shown in Figure.. Now, using p s n pproimtion of f on this suintervl, ou hve, Theorem.7, f d p d p p n n p p p f f f. p Repeting this procedure on the entire intervl, produces the following theorem. THEOREM. Simpson s Rule (n is even) Let f e continuous on,. Simpson s Rule for pproimting f d is f d n f f f f... f n f n. Moreover, s n, the righthnd side pproches f d. NOTE Oserve tht the coefficients in Simpson s Rule hve the following pttern.... In Emple, the Trpezoidl Rule ws used to estimte emple, Simpson s Rule is pplied to the sme integrl. sin d. In the net EXAMPLE Approimtion with Simpson s Rule NOTE In Emple, the Trpezoidl Rule with n pproimted sin d s.97. In Emple, Simpson s Rule with n gve n pproimtion of.. The ntiderivtive would produce the true vlue of. Use Simpson s Rule to pproimte Compre the results for n nd n. Solution When n, ou hve sin d. sin d sin sin sin When n, ou hve sin d.. sin sin.5.
5 _.qd // : PM Pge SECTION. Numericl Integrtion Error Anlsis If ou must use n pproimtion technique, it is importnt to know how ccurte ou cn epect the pproimtion to e. The following theorem, which is listed without proof, gives the formuls for estimting the errors involved in the use of Simpson s Rule nd the Trpezoidl Rule. THEOREM.9 Errors in the Trpezoidl Rule nd Simpson s Rule If f hs continuous second derivtive on,, then the error E in pproimting f d the Trpezoidl Rule is E n m f, Trpezoidl Rule Moreover, if f hs continuous fourth derivtive on,, then the error E in pproimting f d Simpson s Rule is. E 5 n m f,. Simpson s Rule TECHNOLOGY If ou hve ccess to computer lger sstem, use it to evlute the definite integrl in Emple. You should otin vlue of d ln.779. ( ln represents the nturl logrithmic function, which ou will stud in Section 5..) = +. d. Figure.5 n = Theorem.9 sttes tht the errors generted the Trpezoidl Rule nd Simpson s Rule hve upper ounds dependent on the etreme vlues of f nd f in the intervl,. Furthermore, these errors cn e mde ritrril smll incresing n, provided tht re continuous nd therefore ounded in,. EXAMPLE The Approimte Error in the Trpezoidl Rule Determine vlue of n such tht the Trpezoidl Rule will pproimte the vlue of d with n error tht is less thn.. Solution Begin letting f nd finding the second derivtive of f. f nd f The mimum vlue of f on the intervl, is f. So, Theorem.9, ou cn write E n f n n. To otin n error E tht is less thn., ou must choose n such tht n. n So, ou cn choose n (ecuse n must e greter thn or equl to.9) nd ppl the Trpezoidl Rule, s shown in Figure.5, to otin d.5. f nd f n.9 So, with n error no lrger thn., ou know tht. d..
6 _.qd // : PM Pge CHAPTER Integrtion In Eercises, use the Trpezoidl Rule nd Simpson s Rule to pproimte the vlue of the definite integrl for the given vlue of n. Round our nswer to four deciml plces nd compre the results with the ect vlue of the definite integrl.. d, n.. d, n. 5. d, n. 7. d, n. 9. d, n. In Eercises, pproimte the definite integrl using the Trpezoidl Rule nd Simpson s Rule with n. Compre these results with the pproimtion of the integrl using grphing utilit.. d.. d. 5. cos d. 7. sin d. 9.. Eercises for Section. 9 In Eercises, use the error formuls in Theorem.9 to estimte the error in pproimting the integrl, with n, using () the Trpezoidl Rule nd () Simpson s Rule.. tn d f d, f sin,,. d. > Writing Aout Concepts d, n d, n d, n d, n d, n d sin d tn d cos d. If the function f is concve upwrd on the intervl,, will the Trpezoidl Rule ield result greter thn or less thn f d? Eplin.. The Trpezoidl Rule nd Simpson s Rule ield pproimtions of definite integrl f d sed on polnomil pproimtions of f. Wht degree polnomil is used for ech? d 5.. d 7. cos d. In Eercises 9, use the error formuls in Theorem.9 to find n such tht the error in the pproimtion of the definite integrl is less thn. using () the Trpezoidl Rule nd () Simpson s Rule. 9.. d. d.. cos d. In Eercises 5, use computer lger sstem nd the error formuls to find n such tht the error in the pproimtion of the definite integrl is less thn. using () the Trpezoidl Rule nd () Simpson s Rule. 5. d. 7. tn d. 9. Approimte the re of the shded region using () the Trpezoidl Rule nd () Simpson s Rule with n. See for workedout solutions to oddnumered eercises. 5 Figure for 9 Figure for. Approimte the re of the shded region using () the Trpezoidl Rule nd () Simpson s Rule with n.. Progrmming Write progrm for grphing utilit to pproimte definite integrl using the Trpezoidl Rule nd Simpson s Rule. Strt with the progrm written in Section., Eercises 59, nd note tht the Trpezoidl Rule cn e written s Tn Ln Rn nd Simpson s Rule cn e written s Sn Tn Mn. [Recll tht Ln, Mn, nd Rn represent the Riemnn sums using the lefthnd endpoints, midpoints, nd righthnd endpoints of suintervls of equl width.] d sin d d d sin d d sin d
7 _.qd // : PM Pge 5 SECTION. Numericl Integrtion 5 Progrmming In Eercises, use the progrm in Eercise to pproimte the definite integrl nd complete the tle. n Ln. d. d. 5. Are Use Simpson s Rule with n to pproimte the re of the region ounded the grphs of cos,,, nd.. Circumference The elliptic integrl sin d gives the circumference of n ellipse. Use Simpson s Rule with n to pproimte the circumference. 7. Work To determine the size of the motor required to operte press, compn must know the mount of work done when the press moves n oject linerl 5 feet. The vrile force to move the oject is F 5 where F is given in pounds nd gives the position of the unit in feet. Use Simpson s Rule with n to pproimte the work W (in footpounds) done through one ccle if 5 W F d.. The tle lists severl mesurements gthered in n eperiment to pproimte n unknown continuous function f. () Approimte the integrl f d using the Trpezoidl Rule nd Simpson s Rule. Mn Rn.5 Tn Sn sin d () Use grphing utilit to find model of the form c d for the dt. Integrte the resulting polnomil over, nd compre the result with prt (). Approimtion of Pi In Eercises 9 nd 5, use Simpson s Rule with n to pproimte using the given eqution. (In Section 5.7, ou will e le to evlute the integrl using inverse trigonometric functions.) d d Are In Eercises 5 nd 5, use the Trpezoidl Rule to estimte the numer of squre meters of lnd in lot where nd re mesured in meters, s shown in the figures. The lnd is ounded strem nd two stright rods tht meet t right ngles Prove tht Simpson s Rule is ect when pproimting the integrl of cuic polnomil function, nd demonstrte the result for d, n. 5. Use Simpson s Rule with n nd computer lger sstem to pproimte t in the integrl eqution t sin d. 55. Prove tht ou cn find polnomil p A B C tht psses through n three points,,,, nd,, where the s re distinct. i 5 5 Rod Rod Strem Strem Rod Rod
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