Approimte Integrtion: Trpezoid Rule nd Simpson s Rule There re two situtions in which it is impossible to find the ect vlue of definite integrl. The first sitution rises from the fct tht in order to evlute b f sd d using the Fun dmentl Theorem of Clculus we need to know n ntiderivtive of f. Sometimes, however, it is difficult, or even impossible, to find n ntiderivtive (see Section 5.7). For emple, it is impossible to evlute the following integrls ectl: e d s d 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning () Left endpoint pproimtion (b) Right endpoint pproimtion (c) Midpoint pproimtion FIGURE The second sitution rises when the function is determined from scientific eperiment through instrument redings or collected dt. There m be no formul for the function (see Emple 5). In both cses we need to find pproimte vlues of definite integrls. We lred know one such method. Recll tht the definite integrl is defined s limit of Riemnn sums, so n Riemnn sum could be used s n pproimtion to the integrl: If we divide f, bg into n subintervls of equl length D sb dn, then we hve b f sd d < o n f s*d i D i where i * is n point in the ith subintervl f i, i g. If i * is chosen to be the left endpoint of the intervl, then i * i nd we hve b f sd d < L n o n f s i d D i If f sd >, then the integrl represents n re nd () represents n pproimtion of this re b the rectngles shown in Figure (). If we choose i * to be the right endpoint, then i * i nd we hve b f sd d < R n o n f s i d D i [See Figure (b).] The pproimtions L n nd R n defined b Equtions nd re clled the left endpoint pproimtion nd right endpoint pproimtion, respectivel. We hve lso considered the cse where i * is chosen to be the midpoint i of the subintervl f i, i g. Figure (c) shows the midpoint pproimtion M n, which ppers to be better thn either L n or R n. Midpoint Rule where nd b f sd d < M n D f f s d f s d f s n dg D b n i s i i d midpoint of f i, i g
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE Another pproimtion, clled the Trpezoidl Rule, results from verging the pproimtions in Equtions nd : b f sd d < Fo n f s i d D o n f s i d DG D Fo n s f s i d f s i ddg i i i D fs f s d f s dd s f s d f s dd s f s n d f s n ddg D f f s d f s d f s d f s n d f s n dg Trpezoidl Rule FIGURE Trpezoidl pproimtion FIGURE = b f sd d < T n D f f s d f s d f s d f s n d f s n dg where D sb dn nd i i D. The reson for the nme Trpezoidl Rule cn be seen from Figure, which illustrtes the cse with f sd > nd n 4. The re of the trpezoid tht lies bove the ith subintervl is D S f s id f s i d D D f f s id f s i dg nd if we dd the res of ll these trpezoids, we get the right side of the Trpezoidl Rule. EXAMPLE Use () the Trpezoidl Rule nd (b) the Midpoint Rule with n 5 to pproimte the integrl sd d. SOLUTION () With n 5,, nd b, we hve D s d5., nd so the Trpezoidl Rule gives d < T 5. f f sd f s.d f s.4d f s.6d f s.8d f sdg.s..4.6.8 D <.69565 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning This pproimtion is illustrted in Figure. = (b) The midpoints of the five subintervls re.,.,.5,.7, nd.9, so the Midpoint Rule gives d < D f f s.d f s.d f s.5d f s.7d f s.9dg S 5...5.7.9D FIGURE 4 <.6998 This pproimtion is illustrted in Figure 4. n
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE In Emple we delibertel chose n integrl whose vlue cn be computed eplicitl so tht we cn see how ccurte the Trpezoidl nd Midpoint Rules re. B the Fundmentl Theorem of Clculus, d ln g ln.6947... b f sd d pproimtion error The error in using n pproimtion is defined to be the mount tht needs to be dded to the pproimtion to mke it ect. From the vlues in Emple we see tht the errors in the Trpezoidl nd Midpoint Rule pproimtions for n 5 re E T <.488 nd E M <.9 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning Approimtions to Corresponding errors d It turns out tht these observtions re true in most cses. C B A B Q A P i- i FIGURE 5 P C i D R D In generl, we hve E T b f sd d T n nd E M b f sd d M n The following tbles show the results of clcultions similr to those in Emple, but for n 5,, nd nd for the left nd right endpoint pproimtions s well s the Trpezoidl nd Midpoint Rules. n L n R n T n M n 5.74565.64565.69565.6998.7877.66877.6977.6985.758.688.69.6969 n E L E R E T E M 5.5488.475.488.9.564.476.64..656.44.56.78 We cn mke severl observtions from these tbles:. In ll of the methods we get more ccurte pproimtions when we increse the vlue of n. (But ver lrge vlues of n result in so mn rithmetic opertions tht we hve to bewre of ccumulted round-off error.). The errors in the left nd right endpoint pproimtions re opposite in sign nd pper to decrese b fctor of bout when we double the vlue of n.. The Trpezoidl nd Midpoint Rules re much more ccurte thn the endpoint pproimtions. 4. The errors in the Trpezoidl nd Midpoint Rules re opposite in sign nd pper to decrese b fctor of bout 4 when we double the vlue of n. 5. The size of the error in the Midpoint Rule is bout hlf the size of the error in the Trpezoidl Rule. Figure 5 shows wh we cn usull epect the Midpoint Rule to be more ccurte thn the Trpezoidl Rule. The re of tpicl rectngle in the Midpoint Rule is the sme s the re of the trpezoid ABCD whose upper side is tngent to the grph t P. The re of this trpezoid is closer to the re under the grph thn is the re of the trpezoid AQRD used in the Trpezoidl Rule. [The midpoint error (shded red) is smller thn the trpezoidl error (shded blue).]
4 APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE These observtions re corroborted in the following error estimtes, which re proved in books on numericl nlsis. Notice tht Observtion 4 corresponds to the n in ech denomintor becuse snd 4n. The fct tht the estimtes depend on the size of the second derivtive is not surprising if ou look t Figure 5, becuse f sd mesures how much the grph is curved. [Recll tht f sd mesures how fst the slope of f sd chnges.] Error Bounds Suppose f sd < K for < < b. If E T nd E M re the errors in the Trpezoidl nd Midpoint Rules, then E T Ksb d < nd n E Ksb d M < 4n K cn be n number lrger thn ll the vlues of f sd, but smller vlues of K give better error bounds. Let s ppl this error estimte to the Trpezoidl Rule pproimtion in Emple. If f sd, then f 9sd nd f sd. Becuse < <, we hve <, so f sd Z Z < Therefore, tking K,, b, nd n 5 in the error estimte (), we see tht E T < s d s5d 5 <.6667 Compring this error estimte of.6667 with the ctul error of bout.488, we see tht it cn hppen tht the ctul error is substntill less thn the upper bound for the error given b (). EXAMPLE How lrge should we tke n in order to gurntee tht the Trpezoidl nd Midpoint Rule pproimtions for sd d re ccurte to within.? SOLUTION We sw in the preceding clcultion tht f sd < for < <, so we cn tke K,, nd b in (). Accurc to within. mens tht the size of the error should be less thn.. Therefore we choose n so tht Solving the inequlit for n, we get n. sd n,. s.d 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning It s quite possible tht lower vlue for n would suffice, but 4 is the smllest vlue for which the error bound formul cn gurntee us ccurc to within.. or n. s.6 < 4.8 Thus n 4 will ensure the desired ccurc. For the sme ccurc with the Midpoint Rule we choose n so tht sd 4n,. nd so n. < 9 n s.
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 5 EXAMPLE () Use the Midpoint Rule with n to pproimte the integrl e d. (b) Give n upper bound for the error involved in this pproimtion. SOLUTION () Since, b, nd n, the Midpoint Rule gives =e e d < D f f s.5d f s.5d f s.85d f s.95dg.fe.5 e.5 e.65 e.5 e.5 e.5 e.45 e.565 e.75 e.95 g 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning FIGURE 6 Error estimtes give upper bounds for the error. The re theoreticl, worstcse scenrios. The ctul error in this cse turns out to be bout.. P P <.469 Figure 6 illustrtes this pproimtion. (b) Since f sd e, we hve f 9sd e nd f sd s 4 de. Also, since < <, we hve < nd so < f sd s 4 de < 6e Tking K 6e,, b, nd n in the error estimte (), we see tht n upper bound for the error is Simpson s Rule 6esd 4sd e 4 <.7 n Another rule for pproimte integrtion results from using prbols insted of stright line segments to pproimte curve. As before, we divide f, bg into n subintervls of equl length h D sb dn, but this time we ssume tht n is n even number. Then on ech consecutive pir of intervls we pproimte the curve f sd > b prbol s shown in Figure 7. If i f s i d, then P i s i, i d is the point on the curve ling bove i. A tpicl prbol psses through three consecutive points P i, P i, nd P i. P Pß P (_h, ) P (, ) P P P P (h, fi) = ß=b _h h FIGURE 7 FIGURE 8 To simplif our clcultions, we first consider the cse where h,, nd h. (See Figure 8.) We know tht the eqution of the prbol through P, P, nd P is of the form A B C nd so the re under the prbol from h
6 APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE to h is Here we hve used Theorem 5.4.6. Notice tht A C is even nd B is odd. h h sa B Cd d h sa Cd d FA C G SA D h Ch h sah 6Cd h But, since the prbol psses through P sh, d, P s, d, nd P sh, d, we hve Ashd Bshd C Ah Bh C C nd therefore Ah Bh C 4 Ah 6C Thus we cn rewrite the re under the prbol s h s 4 d Now b shifting this prbol horizontll we do not chnge the re under it. This mens tht the re under the prbol through P, P, nd P from to in Figure 7 is still h s 4 d Similrl, the re under the prbol through P, P, nd P 4 from to 4 is h s 4 4 d If we compute the res under ll the prbols in this mnner nd dd the results, we get b f sd d < h s 4 d h s 4 4 d h s n 4 n n d h s 4 4 4 n 4 n n d Although we hve derived this pproimtion for the cse in which f sd >, it is resonble pproimtion for n continuous function f nd is clled Simpson s Rule fter the English mthemticin Thoms Simpson (7 76). Note the pttern of coefficients:, 4,, 4,, 4,,..., 4,, 4,. 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning Simpson Thoms Simpson ws wever who tught himself mthemtics nd went on to become one of the best English mthemticins of the 8th centur. Wht we cll Simpson s Rule ws ctull known to Cvlieri nd Gregor in the 7th centur, but Simpson populrized it in his book Mthemticl Disserttions (74). Simpson s Rule b f sd d < S n D f f s d 4 f s d f s d 4 f s d where n is even nd D sb dn. f s n d 4 f s n d f s n dg
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 7 EXAMPLE 4 Use Simpson s Rule with n to pproimte sd d. SOLUTION Putting f sd, n, nd D. in Simpson s Rule, we obtin d < S D f f sd 4 f s.d f s.d 4 f s.d f s.8d 4 f s.9d f sdg S. 4.. 4..4 4.5.6 4.7.8 4.9 D <.695 n 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning Notice tht, in Emple 4, Simpson s Rule gives us much better pproimtion ss <.695d to the true vlue of the integrl sln <.6947...d thn does the Trpezoidl Rule st <.6977d or the Midpoint Rule sm <.6985d. It turns out (see Eercise 5) tht the pproimtions in Simpson s Rule re weighted verges of those in the Trpezoidl nd Midpoint Rules: S n T n M n (Recll tht E T nd E M usull hve opposite signs nd E M is bout hlf the size of E T.) In mn pplictions of clculus we need to evlute n integrl even if no eplicit formul is known for s function of. A function m be given grphicll or s tble of vlues of collected dt. If there is evidence tht the vlues re not chnging rpidl, then the Trpezoidl Rule or Simpson s Rule cn still be used to find n pproimte vlue for b d, the integrl of with respect to. EXAMPLE 5 Figure 9 shows dt trffic on the link from the United Sttes to SWITCH, the Swiss cdemic nd reserch network, on Februr, 998. Dstd is the dt throughput, mesured in megbits per second smbsd. Use Simpson s Rule to estimte the totl mount of dt trnsmitted on the link from midnight to noon on tht d. D 8 6 4 FIGURE 9 6 9 5 8 4 t (hours) SOLUTION Becuse we wnt the units to be consistent nd Dstd is mesured in meg bits per second, we convert the units for t from hours to seconds. If we let Astd be the mount of dt (in megbits) trnsmitted b time t, where t is mesured in seconds, then A9std Dstd. So, b the Net Chnge Theorem (see Section 5.4), the totl mount
8 APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE of dt trnsmitted b noon (when t 6 4,) is As4,d 4, Dstd dt We estimte the vlues of Dstd t hourl intervls from the grph nd compile them in the tble. n M n S n 4.69989.69545 8.696655.694765 6.695.6947 n E M E S 4.979.75 8.4866.47 6.97. t (hours) t (seconds) Dstd t (hours) t (seconds) Dstd. 7 5,.,6.7 8 8,8.8 7,.9 9,4 5.7,8.7 6, 7. 4 4,4. 9,6 7.7 5 8,. 4, 7.9 6,6. Then we use Simpson s Rule with n nd Dt 6 to estimte the integrl: 4, Astd dt < Dt < 6 4,88 fdsd 4Ds6d Ds7d 4Ds9,6d Ds4,dg f. 4s.7d s.9d 4s.7d s.d 4s.d s.d 4s.d s.8d 4s5.7d s7.d 4s7.7d 7.9g Thus the totl mount of dt trnsmitted from midnight to noon is bout 44, megb its, or 44 gigbits. n The tble in the mrgin shows how Simpson s Rule compres with the Midpoint Rule for the integrl sd d, whose vlue is bout.69478. The second tble shows how the error E S in Simpson s Rule decreses b fctor of bout 6 when n is doubled. (In Eercises 7 nd 8 ou re sked to verif this for two dditionl integrls.) Tht is consistent with the ppernce of n 4 in the denomintor of the following error estimte for Simpson s Rule. It is similr to the estimtes given in () for the Trpezoidl nd Midpoint Rules, but it uses the fourth derivtive of f. 4 Error Bound for Simpson s Rule Suppose tht f s4d sd If E S is the error involved in using Simpson s Rule, then E S < Ksb d5 8n 4 < K for < < b. 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning EXAMPLE 6 How lrge should we tke n in order to gurntee tht the Simpson s Rule pproimtion for sd d is ccurte to within.?
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 9 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning Mn clcultors nd computer lgebr sstems hve built-in lgorithm tht computes n pproimtion of definite integrl. Some of these mchines use Simpson s Rule; others use more sophisticted techniques such s dptive numericl integrtion. This mens tht if function fluctutes much more on certin prt of the intervl thn it does elsewhere, then tht prt gets divided into more sub intervls. This strteg reduces the number of clcultions required to chieve prescribed ccurc. Figure illustrtes the clcultion in Emple 7. Notice tht the prbolic rcs re so close to the grph of e tht the re prcticll indistinguishble from it. =e SOLUTION If f sd, then f s4d sd 4 5. Since >, we hve < nd so < 4 f s4d sd Z 4 5 Z Therefore we cn tke K 4 in (4). Thus, for n error less thn., we should choose n so tht This gives n 4. or n. 4sd 5 8n 4,. 4 8s.d s 4.75 < 6.4 Therefore n 8 (n must be even) gives the desired ccurc. (Compre this with Emple, where we obtined n 4 for the Trpezoidl Rule nd n 9 for the Midpoint Rule.) n EXAMPLE 7 () Use Simpson s Rule with n to pproimte the integrl e d. (b) Estimte the error involved in this pproimtion. SOLUTION () If n, then D. nd Simpson s Rule gives e d < D f f sd 4 f s.d f s.d f s.8d 4 f s.9d f sdg. fe 4e. e.4 4e.9 e.6 4e.5 e.6 <.4668 4e.49 e.64 4e.8 e g (b) The fourth derivtive of f sd e is nd so, since < <, we hve f s4d sd s 48 6 4 de < f s4d sd < s 48 6de 76e FIGURE Therefore, putting K 76e,, b, nd n in (4), we see tht the error is t most 76esd 5 8sd 4 <.5 (Compre this with Emple.) Thus, correct to three deciml plces, we hve e d <.46 n
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE Eercises ; ;. Let I 4 f sd d, where f is the function whose grph is shown. () Use the grph to find L, R, nd M. (b) Are these underestimtes or overestimtes of I? (c) Use the grph to find T. How does it compre with I? (d) For n vlue of n, list the numbers L n, R n, M n, T n, nd I in incresing order. f 4. The left, right, Trpezoidl, nd Midpoint Rule pproimtions were used to estimte f sd d, where f is the function whose grph is shown. The estimtes were.78,.8675,.86, nd.954, nd the sme number of subintervls were used in ech cse. () Which rule produced which estimte? (b) Between which two pproimtions does the true vlue of f sd d lie? =ƒ. Estimte coss d d using () the Trpezoidl Rule nd (b) the Midpoint Rule, ech with n 4. From grph of the integrnd, decide whether our nswers re underestimtes or overestimtes. Wht cn ou conclude bout the true vlue of the integrl? 4. Drw the grph of f sd sin( ) in the viewing rectngle f, g b f,.5g nd let I f sd d. () Use the grph to decide whether L, R, M, nd T underestimte or overestimte I. (b) For n vlue of n, list the numbers L n, R n, M n, T n, nd I in incresing order. (c) Compute L 5, R 5, M 5, nd T 5. From the grph, which do ou think gives the best estimte of I? 5 6 Use () the Midpoint Rule nd (b) Simpson s Rule to pproimte the given integrl with the specified vlue of n. (Round our nswers to si deciml plces.) Compre our results to the ctul vlue to determine the error in ech pproimtion. 5. d, n 6. cos d, n 4 CAS 7 8 Use () the Trpezoidl Rule, (b) the Midpoint Rule, nd (c) Simpson s Rule to pproimte the given integrl with the specified vlue of n. (Round our nswers to si deciml plces.) 7. s d, n 8. d, n 8 6 9. e d, n. s cos d, n 4. 4 sln d, n 6. sins d d, n. 4 e st sin t dt, n 8 4. sz ez dz, n 5. 5 cos d, n 8 6. 6 lns d d, n 7. e e d, n 8. 4 cos s d, n 9. () Find the pproimtions T 8 nd M 8 for the integrl cos s d d. (b) Estimte the errors in the pproimtions of prt (). (c) How lrge do we hve to choose n so tht the pproimtions T n nd M n to the integrl in prt () re ccurte to within.?. () Find the pproimtions T nd M for e d. (b) Estimte the errors in the pproimtions of prt (). (c) How lrge do we hve to choose n so tht the pproimtions T n nd M n to the integrl in prt () re ccurte to within.?. () Find the pproimtions T, M, nd S for sin d nd the corresponding errors E T, E M, nd E S. (b) Compre the ctul errors in prt () with the error estimtes given b () nd (4). (c) How lrge do we hve to choose n so tht the pproimtions T n, M n, nd S n to the integrl in prt () re ccurte to within.?. How lrge should n be to gurntee tht the Simpson s Rule pproimtion to e d is ccurte to within.?. The trouble with the error estimtes is tht it is often ver difficult to compute four derivtives nd obtin good upper bound K for f s4d sd b hnd. But computer lgebr sstems hve no problem computing f s4d nd grphing it, so we cn esil find vlue for K from mchine grph. This eercise dels with pproimtions to the integrl I f sd d, where f sd e cos. () Use grph to get good upper bound for f sd. (b) Use M to pproimte I. 4 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE CAS (c) Use prt () to estimte the error in prt (b). (d) Use the built-in numericl integrtion cpbilit of our CAS to pproimte I. (e) How does the ctul error compre with the error estimte in prt (c)? (f) Use grph to get good upper bound for f s4d sd. (g) Use S to pproimte I. (h) Use prt (f) to estimte the error in prt (g). (i) How does the ctul error compre with the error estimte in prt (h)? ( j) How lrge should n be to gurntee tht the size of the error in using S n is less thn.? 4. Repet Eercise for the integrl s4 d.. () Use the Midpoint Rule nd the given dt to estimte the vlue of the integrl 5 f sd d. f sd f sd..4.5 4..5.9 4. 4... 4.5.9.5.6 5..5..8 (b) If it is known tht < f sd < for ll, estimte the error involved in the pproimtion in prt (). 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning 5 6 Find the pproimtions L n, R n, T n, nd M n for n 5,, nd. Then compute the corresponding errors E L, E R, E T, nd E M. (Round our nswers to si deciml plces. You m wish to use the sum commnd on computer lgebr sstem.) Wht observtions cn ou mke? In prticulr, wht hppens to the errors when n is doubled? 5. e d 6. d 7 8 Find the pproimtions T n, M n, nd S n for n 6 nd. Then compute the corresponding errors E T, E M, nd E S. (Round our nswers to si deciml plces. You m wish to use the sum commnd on computer lgebr sstem.) Wht observtions cn ou mke? In prticulr, wht hppens to the errors when n is doubled? 7. 4 d 8. 4 s d 9. Estimte the re under the grph in the figure b using () the Trpezoidl Rule, (b) the Midpoint Rule, nd (c) Simpson s Rule, ech with n 6. 4 5 6. The widths (in meters) of kidne-shped swimming pool were mesured t -meter intervls s indicted in the figure. Use Simpson s Rule to estimte the re of the pool. 6. 5. 7. 6.8 5.6 4.8 4.8. () A tble of vlues of function t is given. Use Simpson s Rule to estimte.6 tsd d. tsd tsd......6..6.4..4..6..6..8.7 (b) If 5 < t s4d sd < for < <.6, estimte the error involved in the pproimtion in prt ().. A grph of the temperture in Boston on August,, is shown. Use Simpson s Rule with n to estimte the verge temperture on tht d. T 8 7 6 (F) 4 8 noon 4 8 t 4. A rdr gun ws used to record the speed of runner during the first 5 seconds of rce (see the tble). Use Simpson s Rule to estimte the distnce the runner covered during those 5 seconds. t (s) v (ms) t (s) v (ms)..5.5 4.67.5.67. 7.4 4..76.5 8.86 4.5.8. 9.7 5..8.5.
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 5. The grph of the ccelertion std of cr mesured in fts is shown. Use Simpson s Rule to estimte the increse in the velocit of the cr during the 6-second time intervl. 8 9. Use Simpson s Rule with n 8 to estimte the volume of the solid obtined b rotting the region shown in the figure bout () the -is nd (b) the -is. 4 4 4 6 8 4 6 t (seconds) 6. Wter leked from tnk t rte of rstd liters per hour, where the grph of r is s shown. Use Simpson s Rule to estimte the totl mount of wter tht leked out during the first 6 hours. r 4 4 6 t (seconds) 7. The tble (supplied b Sn Diego Gs nd Electric) gives the power consumption P in megwtts in Sn Diego Count from midnight to 6: m on d in December. Use Simpson s Rule to estimte the energ used during tht time period. (Use the fct tht power is the derivtive of energ.) t P t P : 84 : 6 : 75 4: 6 : 686 4: 666 : 646 5: 745 : 67 5: 886 : 69 6: 5 : 64 8. Shown is the grph of trffic on n Internet service provider s T dt line from midnight to 8: m. D is the dt throughput, mesured in megbits per second. Use Simpson s Rule to estimte the totl mount of dt trnsmitted during tht time period. D.8.4 4 6 8 t (hours) CAS 4. The tble shows vlues of force function f sd, where is mesured in meters nd f sd in newtons. Use Simpson s Rule to estimte the work done b the force in moving n object distnce of 8 m. 6 9 5 8 f sd 9.8 9. 8.5 8. 7.7 7.5 7.4 4. The region bounded b the curve s e d, the - nd -es, nd the line is rotted bout the -is. Use Simpson s Rule with n to estimte the volume of the resulting solid. 4. The figure shows pendulum with length L tht mkes mimum ngle with the verticl. Using Newton s Second Lw, it cn be shown tht the period T (the time for one complete swing) is given b T 4Î L t d s k sin where k sin( ) nd t is the ccelertion due to grvit. If L m nd 4, use Simpson s Rule with n to find the period. 4. The intensit of light with wvelength trveling through diffrction grting with N slits t n ngle is given b Isd N sin kk, where k snd sin d nd d is the distnce between djcent slits. A helium-neon lser with wvelength 6.8 9 m is emitting nrrow bnd of light, given b 6,, 6, through grting with, slits spced 4 m prt. Use the Midpoint Rule with n to estimte the totl light intensit 6 6 Isd d emerging from the grting. 44. Use the Trpezoidl Rule with n to pproimte cossd d. Compre our result to the ctul vlue. Cn ou eplin the discrepnc? 45. Sketch the grph of continuous function on f, g for which the Trpezoidl Rule with n is more ccurte thn the Midpoint Rule. 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning
6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 46. Sketch the grph of continuous function on f, g for which the right endpoint pproimtion with n is more ccurte thn Simpson s Rule. 47. If f is positive function nd f sd, for < < b, show tht T n, b f sd d, M n 48. Show tht if f is polnomil of degree or lower, then Simpson s Rule gives the ect vlue of b f sd d. 49. Show tht stn Mnd Tn. 5. Show tht Tn Mn Sn.
4 APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE Answers. () L 6, R, M < 9.6 (b) L is n underestimte, R nd M re overestimtes. (c) T 9, I (d) L n, T n, I, M n, R n. () T 4 <.895759 (underestimte) (b) M 4 <.9897 (overestimte); T 4, I, M 4 5. () M <.86598, E M <.879 (b) S <.84779, E S <.6 7. ().566 (b).586 (c).559 9. ().668 (b).66477 (c).6644. ().594 (b).6846 (c).6976. () 4.568 (b) 4.74856 (c) 4.675 5. ().495 (b).54 (c).56 7. () 8.685 (b) 8.698 (c) 8.54 9. () T 8 <.9, M 8 <.956 (b) E T <.78, E M <.9 (c) n 7 for T n, n 5 for M n. () T <.9854, E T <.6476; M <.848, E M <.848; S <., E S <. (b) ET <.589, EM <.99, ES <.7 (c) n 59 for T n, n 6 for M n, n for S n. ().8 (b) 7.9549658 (c).894 (d) 7.954965 (e) The ctul error is much smller. (f).9 (g) 7.957894 (h).59 (i) The ctul error is smller. (j) n > 5 5. n L n R n T n M n 5.7494.86599.477.996.86778.96.696.9985.9967.6888.94.99958 n E L E R E T E M 5.5757.86599.477.779.8.96.696.848.67.6888.94.46 Observtions re the sme s fter Emple. 7. n T n M n S n 6 6.69547 6.557 6.49 6.474 6.68 6.46 n E T E M E S 6.9547.4748.9.74.699.6 Observtions re the sme s fter Emple. 9. () 9 (b) 8.6 (c) 8.6. () 4.4 (b). 7.8 F 5. 7.7 fts 7.,77 megwtt-hours 9. () 9 (b) 88 4. 8 4. 59.4 45..5.5 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 5 Solutions. () =( ) =(4 ) = = ( ) = ( ) +( ) =[() + ()] = (5+5) = 6 = = () = ( ) +( ) =[() + (4)] = (5+5) = = = ( ) = ( ) +( ) =[() + ()] (6+) = 96 = 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning (b) is n underestimte, since the re under the smll rectngles is less thn the re under the curve, nd is n overestimte, since the re under the lrge rectngles is greter thn the re under the curve. It ppers tht is n overestimte, though it is firl close to. See the solution to Eercise 47 for proof of the fct tht if is concve down on [],then the Midpoint Rule is n overestimte of (). (c) = [( )+( )+( )] = [() + () + (4)] = 5+(5) + 5 =9. This pproimtion is n underestimte, since the grph is concve down. Thus, =9. See the solution to Eercise 47 for generl proof of this conclusion. (d) For n,wewillhve.. () =cos, = 4 = 4 () 4 = 4 () + 4 + 4 + 4 + () 895759 (b) 4 = 4 8 + 8 + 5 8 + 7 8 9897 The grph shows tht is concve down on [ ]. So 4 is n underestimte nd 4 is n overestimte. We cn conclude tht 895759 cos 9897. 5. () ()= +, = = = 5 = 5 + + 5 + + 9 86598 (b) = 5 () + 4 5 + 5 +4 5 + 4 5 + +4 9 5 + () 84779 Actul: = + = ln + [ =+, =] = ln 5 ln = ln 5 8479 Errors: = ctul = 879 = ctul = 6
6 APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 7. () =, = = = () = [() + () + () + () + (4) + (5) +(6) + (7) + (8) + (9) + ()] 566 (b) = [(5) + (5) + (5) + (5) + (45) + (55) + (65) + (75) + (85) + (95)] 586 (c) = [() + 4() + () + 4() + (4) +4(5) + (6) + 4(7) + (8) + 4(9) + ()] 559 9. () = +, = = = 5 () = 5 [() + () + (4) + (6) + (8) + () +() + (4) + (6) + (8) + ()] 668 (b) = [() + () + (5) + (7) + (9) + () + () + (5) + (7) + (9)] 5 66477 (c) = [() + 4() + (4) + 4(6) + (8) 5 +4() + () + 4(4) + (6) + 4(8) + ()] 6644. () = ln, = 4 6 = () 6 = [() + (5) + () + (5) + () + (5) + (4)] 594 (b) 6 = [(5) + (75) + (5) + (75) + (5) + (75)] 6846 (c) 6 = [() + 4(5) + () + 4(5) + () + 4(5) + (4)] 6976. () = sin, = 4 8 = () 8 = () + +() + +() + 5 +() + 7 + (4) 4568 (b) 8 = 4 + 4 + 5 4 + 7 4 + 9 4 + 4 + 4 + 5 4 474856 (c) 8 = () + 4 +() + 4 +() + 4 5 +() + 4 7 + (4) 4675 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning 5. () = cos, = 5 8 = () 8 = () + +() + +(4) + 9 + (5) 495 (b) 8 = 5 4 + 7 4 + 9 4 + 4 + 4 + 5 4 + 7 4 + 9 4 54 (c) 8 = () + 4 +() + 4 5 +() + 4 7 +(4) + 4 9 + (5) 56
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 7 7. () =, = ( ) = 5 () = [( ) + ( 8) + ( 6) + ( 4) + ( ) + () 5 +() + (4) + (6) + (8) + ()] 8685 (b) = [( 9) + ( 7) + ( 5) + ( ) + ( ) + () + () + (5) + (7) + (9)] 5 8698 (c) = 5 [( ) + 4( 8) + ( 6) + 4( 4) + ( ) +4() + () + 4(4) + (6) + 4(8) + ()] 854 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning 9. () =cos( ), = 8 = 8 () 8 = 8 () + 8 + 8 + + 7 8 + () 9 8 = 8 6 + 6 + 5 6 + + 5 6 =956 (b) () =cos( ), () = sin( ), () = sin( ) 4 cos( ).For, sin nd cos re positive, so () =sin( )+4 cos( ) +4 =6since sin( ) nd cos for ll, nd for. Sofor =8,wetke =6, =,nd =in Theorem, to get 6 ( 8 )= 8 =785 nd 56 =965. [A better estimte is obtined b noting from grph of tht () 4 for.] (c) Tke =6[s in prt (b)] in Theorem. ( ) 6( ) 4 4 5 7. Tke =7for.For, gin tke =6in 4 Theorem to get 4 4 4 5 5. Tke =5for.. () =sin, = = () = () + = = + () + 4 + + 5 + + + 9 + () 9854 + + 9 848 +4 + +4 9 + () Since = sin = cos = ( ) =, = 6476, = 848, nd =. (b) () =sin () (),sotke =forllerrorestimtes. ( ) ( ) = = () 589. = 4 99. ( )5 ( )5 = 8 4 8() = 5 4,8, 7. The ctul error is bout 64% of the error estimte in ll three cses.
8 APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE (c) 5 5 4 5 5 4 5 8 4 5 5 4 5 8 Tke =for (since must be even). 58. Tke = 59 for. 594. Tke = 6 for... () Using CAS, we differentite () = cos twice, nd find tht () = cos (sin cos ). From the grph, we see tht the mimum vlue of () occurs t the endpoints of the intervl [ ]. Since () =,wecnuse = or =8. (b) A CAS gives 79549658. (InMple,useStudent[Clculus][RiemnnSum] or Student[Clculus][ApproimteInt].) (c) Using Theorem for the Midpoint Rule, with =,weget 8( ) With =8,weget =89996. 4 (d) A CAS gives 7954965. (e) The ctul error is onl bout 9,muchlessthntheestimteinprt(c). (f) We use the CAS to differentite twice more, nd then grph (4) () = cos (sin 4 6sin cos + 7sin +cos). From the grph, we see tht the mimum vlue of (4) () occurs t the endpoints of the intervl [ ]. Since (4) () = 4, we cn use =4 or =9. ( ) 4 8945995. (g) A CAS gives 7957894. (In Mple, use Student[Clculus][ApproimteInt].) (h) Using Theorem 4 with =4,weget 4( )5 8 4 59568. 9( )5 With =9,weget 599984. 8 4 (i) The ctul error is bout 7954965 7957894 4. This is quite bit smller thn the estimte in prt (h), though the difference is not nerl s gret s it ws in the cse of the Midpoint Rule. 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning ( j) To ensure tht, weusetheorem4: 4()5 4()5 8 4 8 4 4 5,95,6 49. Sowemusttke 5 to ensure tht. ( =9leds to the sme vlue of.)
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 9 5. = =[( ) ] [b prts] = ( ) =, () =, = =5: 5 = [() + () + (4) + (6) + (8)] 7494 5 5 = 5 [() + (4) + (6) + (8) + ()] 86599 5 = [() + () + (4) + (6) + (8) + ()] 477 5 5 = [() + () + (5) + (7) + (9)] 996 5 = 5 7494 = 5757 86599 = 86599 477 = 477 996 = 779 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning =: = [() + () + () + + (9)] 86778 = [() + () + + (9) + ()] 96 = {() + [() + () + + (9)] + ()} 696 = [(5) + (5) + + (85) + (95)] 9985 = 86778 = 8 96 = 96 696 = 696 9985 = 848 =: = [() + (5) + () + + (95)] 9967 = [(5) + () + + (95) + ()] 6888 = {() + [(5) + () + + (95)] + ()} 94 = [(5) + (75) + (5) + + (975)] 99958 = 9967 = 67 6888 = 6888 94 = 94 99958 = 46 5 7494 86599 477 996 86778 96 696 9985 9967 6888 94 99958 5 5757 86599 477 779 8 96 696 848 67 6888 94 46 Observtions:. nd re lws opposite in sign, s re nd.. As is doubled, nd re decresed b bout fctor of,nd nd re decresed b fctor of bout 4.. The Midpoint pproimtion is bout twice s ccurte s the Trpezoidl pproimtion. 4. All the pproimtions become more ccurte s the vlue of increses. 5. The Midpoint nd Trpezoidl pproimtions re much more ccurte thn the endpoint pproimtions.
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 7. = 4 = 5 = =64, () 5 5 =4, = = =6: 6 = 6 () + + + + 4 + 5 + () 669547 6 = 6 6 + 6 + 5 6 + 7 6 + 9 6 + 6 6557 6 = 6 () + 4 + +4 + 4 +4 5 + () 649 = 6 64 669547 = 9547 64 6557 = 4748 64 649 = 9 =: = () + 6 + 6 + 6 + + 6 + () 6474 6 = + + 5 + + 668 6 = () + 4 6 + 6 +4 6 + 4 6 + +4 6 + () 646 = 64 6474 = 74 64 668 = 699 64 646 = 6 6 669547 6557 649 6474 668 646 6 9547 4748 9 74 699 6 Observtions:. nd re opposite in sign nd decrese b fctor of bout 4 s is doubled.. The Simpson s pproimtion is much more ccurte thn the Midpoint nd Trpezoidl pproimtions, nd seems to decrese b fctor of bout 6 s is doubled. 9. =( ) =(6 )6 = () 6 = [() + () + () + () + (4) + (5) + (6)] [+(5)+(4)+()+(8) + (4) + ] = (96) = 98 (b) 6 = [(5) + (5) + (5) + (5) + (45) + (55)] [45+47+6++4+] =6 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning (c) 6 = [() + 4() + () + 4() + (4) + 4(5) + (6)] [ + 4(5) + (4) + 4() + (8) + 4(4) + ] = (66) = 5. () 5 () 4 = 5 [(5) + (5) + (5) + (45)] = (9+6+4+9) = 44 4
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE (b) () () =,since (). The error estimte for the Midpoint Rule is ( ) (5 ) = = 4 4(4).. ve = 4 4 () 4 = 4 4 () [ () + 4 () + (4) + 4 (6) + (8) + 4 () + () +4 (4) + (6) + 4 (8) + () + 4 () + (4)] The verge temperture ws bout 644 F. [67 + 4(65) + (6) + 4(58) + (56) + 4(6) + (6) + 4(68) 6 + (7) + 4(69) + (67) + 4(66) + 64] = 6 (7) = 646 F. 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning 5. B the Net Chnge Theorem, the increse in velocit is equl to 6 (). We use Simpson s Rule with =6nd =(6 )6 =to estimte this integrl: 6 () 6 = [() + 4() + () + 4() + (4) + 4(5) + (6)] [ + 4(5) + (4) + 4(98) + (9) + 4(95) + ] = () = 77 fts 7. B the Net Chnge Theorem, the energ used is equl to 6 (). We use Simpson s Rule with =nd = 6 = to estimte this integrl: 6 () = [ () + 4 (5) + () + 4 (5) + () + 4 (5) + () +4 (5) + (4) + 4 (45) + (5) + 4 (55) + (6)] = [84 + 4(75) + (686) + 4(646) + (67) + 4(69) + (64) 6 + 4(6) + (6) + 4(666) + (745) + 4(886) + 5] = (6,64) =,77 megwtt-hours 6 9. () Let = () denote the curve. Using disks, = [()] = () =. Now use Simpson s Rule to pproimte : 8 = [() + 4() + (4) + 4(5) + (6) + 4(7) + (8)] (8) [ +4(5) +(9) +4() +() +4(8) +(4) +4() + ] = (878) Thus, (878) 94 or 9 cubic units. (b) Using clindricl shells, = () = () =. Now use Simpson s Rule to pproimte : 8 = [() + 4 () + 4(4) + 4 5(5) + 6(6) (8) +4 7(7) + 8(8) + 4 9(9) + ()] [() + (5) + 8(9) + () + () + 8(8) + 6(4) + 6() + ()] = (95) Thus, (878) 94 or 9 cubic units.
APPROXIMATE INTEGRATION: TRAPEZOID RULE AND SIMPSON S RULE 4. Using disks, = 5 ( ) = 5 =. Now use Simpson s Rule with () = to pproimte. 8 = 5 (8) [() + 4(5) + () + 4(5) + () + 4(5) + (4) + 4(45) + (5)] 6 (4566) Thus, (4566) 6 cubic units. 6 4. () = sin,where = sin, =,, = 4,nd = 68 9.So() = (4 ) sin, where = (4 )( 4 )sin.now =nd = 6 ( 6 ) = 7,so 68 9 = 7 [( 9) + ( 7) + + (9)] 594. 45. Consider the function whose grph is shown. The re () is close to. The Trpezoidl Rule gives = [() + () + ()] = [ + +]=. The Midpoint Rule gives = [(5) + (5)] = [ + ] =, so the Trpezoidl Rule is more ccurte. 47. Since the Trpezoidl nd Midpoint pproimtions on the intervl [] re the sums of the Trpezoidl nd Midpoint pproimtions on the subintervls [ ], =, we cn focus our ttention on one such intervl. The condition () for mens tht the grph of is concve down s in Figure 5. In tht figure, is the re of the trpezoid, () isthereoftheregion, nd is the re of the trpezoid, so (). In generl, the condition implies tht the grph of on [] lies bove the chord joining the points (()) nd (()). Thus, ().Since is the re under tngent to the grph, nd since implies tht the tngent lies bove the grph, we lso hve (). Thus, (). 49. = [( )+( )+ +( )+( )] nd = [( )+( )+ + ( )+( )],where = ( + ).Now = [( )+( )+( )+( )+( )+ +( )+( )+( )+( )] so ( + )= + = 4 [( )+( )+ +( )+( )] + 4 [( )+( )+ +( )+( )] 6 Cengge Lerning. All Rights Reserved. This content is not et finl nd Cengge Lerning =