4.6 Numerical Integration


 Horace Merritt
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1 .6 Numericl Integrtion 5.6 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. f = = The re of the region cn e pproimted using four trpezoids. Figure. f( ) n f( ) The re of the first trpezoid is f f n. Figure. 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, then while the Fundmentl Theorem of Clculus is still true, it cnnot e esil pplied. In this cse, it is esier to 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 net theorem. cos,, sin Copright Cengge Lerning. All Rights Reserved. M not e copied, scnned, or duplicted, in whole or in prt. Due to electronic rights, some third prt content m e suppressed from the ebook nd/or echpter(s). Editoril review hs deemed tht n suppressed content does not mterill ffect the overll lerning eperience. Cengge Lerning reserves the right to remove dditionl content t n time if susequent rights restrictions require it.
2 6 Chpter Integrtion THEOREM.7 The Trpezoidl Rule Let f e continuous on,. The Trpezoidl Rule for pproimting f d is f d n f f f... f n f n. Moreover, s n, the righthnd side pproches f d. = sin Four suintervls = sin 5 7 Eight suintervls Trpezoidl pproimtions Figure. REMARK pttern. Use the Trpezoidl Rule to pproimte Approimtion with the Trpezoidl Rule Compre the results for n nd n, s shown in Figure.. Solution When n,, nd ou otin When n,, nd ou otin sin d. sin d Oserve tht the coefficients in the Trpezoidl Rule hve the following sin d sin sin sin 6 sin sin sin sin sin 5 7 sin sin sin sin For this prticulr integrl, ou could hve found n ntiderivtive nd determined tht the ect re of the region is. sin sin 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? Copright Cengge Lerning. All Rights Reserved. M not e copied, scnned, or duplicted, in whole or in prt. Due to electronic rights, some third prt content m e suppressed from the ebook nd/or echpter(s). Editoril review hs deemed tht n suppressed content does not mterill ffect the overll lerning eperience. Cengge Lerning reserves the right to remove dditionl content t n time if susequent rights restrictions require it.
3 It is interesting to compre the Trpezoidl Rule with the Midpoint Rule given in Section.. 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 Midpoint Rule Trpezoidl Rule 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, when n 6, 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 sin d ecuse sin hs no elementr ntiderivtive. Yet, the Trpezoidl Rule cn e pplied to estimte this integrl. Simpson s Rule f i i i f i f i.6 Numericl Integrtion 7 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 76), ou tke this procedure one step further nd pproimte f seconddegree polnomils. Before presenting Simpson s Rule, consider the net theorem for evluting integrls of polnomils of degree (or less). THEOREM. If p A B C, then Integrl of p A B C p d 6 p p p. Proof p d A B C d B epnsion nd collection of terms, the epression inside the rckets ecomes p nd ou cn write A A B C B 6 A B 6C A B C A B C A B C p C p d 6 p p p. See LrsonClculus.com for Bruce Edwrds s video of this proof. p Copright Cengge Lerning. All Rights Reserved. M not e copied, scnned, or duplicted, in whole or in prt. Due to electronic rights, some third prt content m e suppressed from the ebook nd/or echpter(s). Editoril review hs deemed tht n suppressed content does not mterill ffect the overll lerning eperience. Cengge Lerning reserves the right to remove dditionl content t n time if susequent rights restrictions require it.
4 Chpter Integrtion p (, ) (, ) (, ) n p d f d Figure.5 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., On ech (doule) suintervl i, i, ou cn pproimte f polnomil p of degree less thn or equl to. (See Eercise 7.) For emple, on the suintervl,, choose the polnomil of lest degree pssing through the points,,,, nd,, s shown in Figure.5. Now, using p s n pproimtion of f on this suintervl, ou hve, Theorem., f d p d 6 p p p n 6 n, p p p f f f. n, n Repeting this procedure on the entire intervl, produces the net theorem. REMARK Oserve tht the coefficients in Simpson s Rule hve the following pttern.... THEOREM.9 Simpson s Rule Let f e continuous on, nd let n e n even integer. 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. REMARK In Section., Emple, the Midpoint Rule with n pproimtes sin d s.5. In Emple, the Trpezoidl Rule with n gives n pproimtion of.96. In Emple, Simpson s Rule with n gives n pproimtion of.5. The ntiderivtive would produce the true vlue of. In Emple, the Trpezoidl Rule ws used to estimte emple, Simpson s Rule is pplied to the sme integrl. Approimtion with Simpson s Rule See LrsonClculus.com for n interctive version of this tpe of emple. 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 d. In the net sin sin.5. Copright Cengge Lerning. All Rights Reserved. M not e copied, scnned, or duplicted, in whole or in prt. Due to electronic rights, some third prt content m e suppressed from the ebook nd/or echpter(s). Editoril review hs deemed tht n suppressed content does not mterill ffect the overll lerning eperience. Cengge Lerning reserves the right to remove dditionl content t n time if susequent rights restrictions require it.
5 .6 Numericl Integrtion 9 FOR FURTHER INFORMATION For proofs of the formuls used for estimting the errors involved in the use of the Midpoint Rule nd Simpson s Rule, see the rticle Elementr Proofs of Error Estimtes for the Midpoint nd Simpson s Rules Edwrd C. Fzeks, Jr. nd Peter R. Mercer in Mthemtics Mgzine. To view this rticle, go to MthArticles.com. Error Anlsis When ou use n pproimtion technique, it is importnt to know how ccurte ou cn epect the pproimtion to e. The net 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. In generl, when using n pproimtion, ou cn think of the error E s the difference etween f d nd the pproimtion. THEOREM. 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. (The smol ln represents the nturl logrithmic function, which ou will stud in Section 5..) = + n =. d.6 Figure.6 Theorem. 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 nd f re continuous nd therefore ounded in,. 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 or equl to.. Solution Begin letting f nd finding the second derivtive of f. f nd f The mimum vlue of on the intervl So, Theorem., ou cn write E To otin n error E tht is less thn., ou must choose n such tht n. n f f n f n n. n.9 So, ou cn choose n (ecuse n must e greter thn or equl to.9) nd ppl the Trpezoidl Rule, s shown in Figure.6, to otin d 6.5., is f. So, dding nd sutrcting the error from this estimte, ou know tht. d.6. Copright Cengge Lerning. All Rights Reserved. M not e copied, scnned, or duplicted, in whole or in prt. Due to electronic rights, some third prt content m e suppressed from the ebook nd/or echpter(s). Editoril review hs deemed tht n suppressed content does not mterill ffect the overll lerning eperience. Cengge Lerning reserves the right to remove dditionl content t n time if susequent rights restrictions require it.
6 Chpter Integrtion.6 Eercises See ClcCht.com for tutoril help nd workedout solutions to oddnumered eercises. Using the Trpezoidl Rule nd Simpson s Rule 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 d, n. 9.. Using the Trpezoidl Rule nd Simpson s Rule 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. sin d cos d d, n. d, n tn d f d, f sin,, > WRITING ABOUT CONCEPTS d, n d, n d, n d, n 6 d sin d tn d sin d. Polnomil Approimtions The Trpezoidl Rule nd Simpson s Rule ield pproimtions of definite integrl f d sed on polnomil pproimtions of f. Wht is the degree of the polnomils used for ech?. Descriing n Error Descrie the size of the error when the Trpezoidl Rule is used to pproimte f d when f is liner function. Use grph to eplin our nswer. Estimting Errors In Eercises 6, use the error formuls in Theorem. to estimte the errors in pproimting the integrl, with n, using () the Trpezoidl Rule nd () Simpson s Rule. 5.. d 5 d d Estimting Errors In Eercises 7, use the error formuls in Theorem. to find n such tht the error in the pproimtion of the definite integrl is less thn or equl to. using () the Trpezoidl Rule nd () Simpson s Rule. 7.. d 9. d. Estimting Errors Using Technolog In Eercises, 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 or equl to. using () the Trpezoidl Rule nd () Simpson s Rule.. d.. tn d. 5. Finding the Are of Region Approimte the re of the shded region using 6 () the Trpezoidl Rule with n. () Simpson s Rule with n. 5 Figure for 5 Figure for 6 6. Finding the Are of Region Approimte the re of the shded region using () the Trpezoidl Rule with n. () Simpson s Rule with n Are Use Simpson s Rule with n to pproimte the re of the region ounded the grphs of cos,,, nd. 6 d cos d d sin d sin d Copright Cengge Lerning. All Rights Reserved. M not e copied, scnned, or duplicted, in whole or in prt. Due to electronic rights, some third prt content m e suppressed from the ebook nd/or echpter(s). Editoril review hs deemed tht n suppressed content does not mterill ffect the overll lerning eperience. Cengge Lerning reserves the right to remove dditionl content t n time if susequent rights restrictions require it.
7 .6 Numericl Integrtion. Circumference The elliptic integrl sin d gives the circumference of n ellipse. Use Simpson s Rule with n to pproimte the circumference. 9. Surveing Use the Trpezoidl Rule to estimte the numer of squre meters of lnd, where nd re mesured in meters, s shown in the figure. The lnd is ounded strem nd two stright rods tht meet t right ngles Rod Strem Rod 6. HOW DO YOU SEE IT? The function f is concve upwrd on the intervl, nd the function g is concve downwrd on the intervl,. 5 f() () Using the Trpezoidl Rule with n, which integrl would e overestimted? Which integrl would e underestimted? Eplin our resoning. () Which rule would ou use for more ccurte pproimtions of nd f d g d, the Trpezoidl Rule or Simpson s Rule? Eplin our resoning. 5 g(). 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 when 5 W F d.. Approimting Function 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. () Use grphing utilit to find model of the form c d for the dt. Integrte the resulting polnomil over, nd compre the result with the integrl from prt (). Approimtion of Pi In Eercises nd, use Simpson s Rule with n 6 to pproimte using the given eqution. (In Section 5.7, ou will e le to evlute the integrl using inverse trigonometric functions.) 6.. d Using Simpson s Rule Use Simpson s Rule with n nd computer lger sstem to pproimte t in the integrl eqution t sin d. 6. Proof Prove tht Simpson s Rule is ect when pproimting the integrl of cuic polnomil function, nd demonstrte the result with n for d. 7. Proof Prove tht ou cn find polnomil p A B C tht psses through n three points,,,, where the s re distinct. i Henrk Sdur/Shutterstock.com d,, nd Copright Cengge Lerning. All Rights Reserved. M not e copied, scnned, or duplicted, in whole or in prt. Due to electronic rights, some third prt content m e suppressed from the ebook nd/or echpter(s). Editoril review hs deemed tht n suppressed content does not mterill ffect the overll lerning eperience. Cengge Lerning reserves the right to remove dditionl content t n time if susequent rights restrictions require it.
8 Answers to OddNumered Eercises A Section.6 (pge ) Trpezoidl Simpson s Ect Trpezoidl Simpson s Grphing Utilit Trpezoidl: Liner (stdegree) polnomils Simpson s: Qudrtic (nddegree) polnomils. ().5 (). 5. () () 7. () n 66 () n 6 9. () n 77 () n. () n () n. () n 6 () n 5. ().5 () ,5 m.,.5 ftl Proof
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