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
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 right-hnd 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.. π π π π Four suintervls = sin π π π π 5π π 7π π Eight suintervls Trpezoidl. pproimtions Figure. 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 Tr It Eplortion A Eplortion B Video TECHNOLOGY Most grphing utilities nd computer lger sstems hve uilt-in 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 uilt-in numericl integrtion progrm to evlute d. Your clcultor should give n error messge. Does ours?
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 first-degree polnomil. In Simpson s Rule, nmed fter the English mthemticin Thoms Simpson (7 7), ou tke this procedure one step further nd pproimte f second-degree 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
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 right-hnd 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. Tr It Eplortion A Open Eplortion
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.. Editle Grph Tr It Eplortion A Eplortion B
CHAPTER Integrtion Eercises for Section. The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem. to view the complete solution of the eercise. to print n enlrged cop of the grph. 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. 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.. 5. d, n. d, n 9 7. d, n. d, n 9.. d, n d, n 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 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. 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 left-hnd endpoints, midpoints, nd right-hnd endpoints of suintervls of equl width.] d sin d d d sin d d sin d
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 foot-pounds) 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...5 7.5 Rn.5 Tn Sn.5.75...5 5.79..5 7..75... 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.) 9. 5. 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. 5. 5. 5 5 9 5 9 95 7 75 9 5 75 7 7 5 9 7 7 9 5 5. 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
CHAPTER Integrtion Review Eercises for Chpter The smol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem. to view the complete solution of the eercise. to print n enlrged cop of the grph. In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, select the MthGrph utton... In Eercises, find the indefinite integrl.. d. 5.. 7.. d d f d sin d 5 cos sec d 9. Find the prticulr solution of the differentil eqution f whose grph psses through the point,.. Find the prticulr solution of the differentil eqution f whose grph psses through the point, nd is tngent to the line 5 t tht point. Slope Fields In Eercises nd, differentil eqution, point, nd slope field re given. () Sketch two pproimte solutions of the differentil eqution on the slope field, one of which psses through the given point. (To print n enlrged cop of the grph, select the MthGrph utton.) () Use integrtion to find the prticulr solution of the differentil eqution nd use grphing utilit to grph the solution. d d.,,., d, d 5 f f. Velocit nd Accelertion An irplne tking off from runw trvels feet efore lifting off. The irplne strts from rest, moves with constnt ccelertion, nd mkes the run in seconds. With wht speed does it lift off?. Velocit nd Accelertion The speed of cr trveling in stright line is reduced from 5 to miles per hour in distnce of feet. Find the distnce in which the cr cn e rought to rest from miles per hour, ssuming the sme constnt decelertion. 5. Velocit nd Accelertion A ll is thrown verticll upwrd from ground level with n initil velocit of 9 feet per second. () How long will it tke the ll to rise to its mimum height? () Wht is the mimum height? (c) When is the velocit of the ll one-hlf the initil velocit? (d) Wht is the height of the ll when its velocit is one-hlf the initil velocit?. Velocit nd Accelertion Repet Eercise 5 for n initil velocit of meters per second. In Eercises 7, use sigm nottion to write the sum. 7.. 9..... n n n n... n n n n... n In Eercises, use the properties of summtion nd Theorem. to evlute the sum. i i. i.. i. 5. Write in sigm nottion () the sum of the first ten positive odd integers, () the sum of the cues of the first n positive integers, nd (c)..... Evlute ech sum for,, 5,, nd 5 7. () () 5 5 i i 5 i (c) (d) 5 i i i i... n n i i ii i 5 i i i 7
REVIEW EXERCISES 7 In Eercises 7 nd, use upper nd lower sums to pproimte the re of the region using the indicted numer of suintervls of equl width. 7.. 9 In Eercises 9, use the limit process to find the re of the region etween the grph of the function nd the -is over the given intervl. Sketch the region.. Use the limit process to find the re of the region ounded 5,,, nd 5.. Consider the region ounded m,,, nd. () Find the upper nd lower sums to pproimte the re of the region when. () Find the upper nd lower sums to pproimte the re of the region when n. (c) Find the re of the region letting n pproch infinit in oth sums in prt (). Show tht in ech cse ou otin the formul for the re of tringle. In Eercises 5 nd, write the limit s definite integrl on the intervl [, ], where is n point in the ith suintervl. 5.. Limit lim lim n i n i In Eercises 7 nd, set up definite integrl tht ields the re of the region. (Do not evlute the integrl.) 5 7 5 9.,,.,,. 5,,.,, c i i c i c i 9 c i i 7. f. f 9 Intervl,, In Eercises 9 nd, sketch the region whose re is given the definite integrl. Then use geometric formul to evlute the integrl. 9. 5 5 d. () (c) f g d. (d) 5f d.. Given f d nd f d, evlute () (c) In Eercises 5, use the Fundmentl Theorem of Clculus to evlute the definite integrl. () () (d). d. 5. t t dt. 7. d. 9. sin d 5. In Eercises 5 5, sketch the grph of the region whose re is given the integrl, nd find the re. 5. d 5. 5. 9 d 5. 55. d 5. In Eercises 57 nd 5, determine the re of the given region. 57. sin 5. cos 5 9. Given f d nd g d, evlute f g d. f d. f d. π d d f d. f g d. f d. t dt 5 d sec t dt d d d π π π
CHAPTER Integrtion In Eercises 59 nd, sketch the region ounded the grphs of the equtions, nd determine its re. () Use integrtion to find the prticulr solution of the differentil eqution nd use grphing utilit to grph the solution. 59..,,, 9 sec,,, d 9., 9. d 9, d d sin,, In Eercises nd, find the verge vlue of the function over the given intervl. Find the vlues of t which the function ssumes its verge vlue, nd grph the function.. f, 9., f,, In Eercises, use the Second Fundmentl Theorem of Clculus to find F... F F t t dt t dt 5. F t. F t dt csc t dt In Eercises 7, find the indefinite integrl. 9. 7. d 7. d 7. 79. sec sec. In Eercises, evlute the definite integrl. Use grphing utilit to verif our result.. d... d 5. d. 7. cos. d 7.. d d 7. sin cos d 7. sin d sin cos 75. 7. cos sin d 77. tn n sec d, n 7. sec tn d d tn d cot csc d d sin d Slope Fields In Eercises 9 nd 9, differentil eqution, point, nd slope field re given. () Sketch two pproimte solutions of the differentil eqution on the slope field, one of which psses through the given point. (To print n enlrged cop of the grph, select the MthGrph utton.) d d 5 d d In Eercises 9 nd 9, find the re of the region. Use grphing utilit to verif our result. 9 9. d 9. 5 9. Fuel Cost Gsoline is incresing in price ccording to the eqution p..t, where p is the dollr price per gllon nd t is the time in ers, with t representing 99. An utomoile is driven 5, miles er nd gets M miles per gllon. The nnul fuel cost is C 5, M Estimte the nnul fuel cost in () nd () 5. 9. Respirtor Ccle After eercising for few minutes, person hs respirtor ccle for which the rte of ir intke is t v.75 sin. Find the volume, in liters, of ir inhled during one ccle integrting the function over the intervl,. In Eercises 95 9, use the Trpezoidl Rule nd Simpson s Rule with n, nd use the integrtion cpilities of grphing utilit, to pproimte the definite integrl. Compre the results. 95. 9. d d 97. cos d 9. sin d 9 9 t t p dt. cos sin d π π
P.S. Prolem Solving 9 P.S. Prolem Solving The smol indictes n eercise in which ou re instructed to use grphing technolog or smolic computer lger sstem. Click on to view the complete solution of the eercise. Click on to print n enlrged cop of the grph.. Let L t dt, >.. The Two-Point Gussin Qudrture Approimtion for f is () Find L. () Find L nd L. (c) Use grphing utilit to pproimte the vlue of (to three deciml plces) for which L. (d) Prove tht L L L for ll positive vlues of nd.. Let F sin t dt. () Use grphing utilit to complete the tle. () Let G F sin t dt. Use grphing utilit to complete the tle nd estimte lim G. (c) Use the definition of the derivtive to find the ect vlue of the limit lim G. In Eercises nd, () write the re under the grph of the given function defined on the given intervl s limit. Then use computer lger sstem to () evlute the sum in prt (), nd (c) evlute the limit using the result of prt ().., Hint: F F G n i nn n n n i., 5, Hint: n i 5 n n n n i 5. The Fresnel function S is defined the integrl S sin t dt...5.9...5.. 5.,.9.95.99.. () Grph the function sin on the intervl,. () Use the grph in prt () to sketch the grph of S on the intervl,. (c) Locte ll reltive etrem of S on the intervl,. (d) Locte ll points of inflection of S on the intervl,. f d f f. () Use this formul to pproimte cos d. Find the error of the pproimtion. () Use this formul to pproimte d. (c) Prove tht the Two-Point Gussin Qudrture Approimtion is ect for ll polnomils of degree or less. 7. Archimedes showed tht the re of prolic rch is equl to the product of the se nd the height (see figure). () Grph the prolic rch ounded 9 nd the -is. Use n pproprite integrl to find the re A. () Find the se nd height of the rch nd verif Archimedes formul. (c) Prove Archimedes formul for generl prol.. Glileo Glilei (5 ) stted the following proposition concerning flling ojects: The time in which n spce is trversed uniforml ccelerting od is equl to the time in which tht sme spce would e trversed the sme od moving t uniform speed whose vlue is the men of the highest speed of the ccelerting od nd the speed just efore ccelertion egn. Use the techniques of this chpter to verif this proposition. 9. The grph of the function f consists of the three line segments joining the points,,,,,, nd,. The function F is defined the integrl F f t dt. () Sketch the grph of f. () Complete the tle. F h 5 7 (c) Find the etrem of F on the intervl,. (d) Determine ll points of inflection of F on the intervl,.
CHAPTER Integrtion. A cr is trveling in stright line for hour. Its velocit v in miles per hour t si-minute intervls is shown in the tle.. Prove. Prove t hours v mi/h t hours v mi/h. Use n pproprite Riemnn sum to evlute the limit. Use n pproprite Riemnn sum to evlute the limit 5. Suppose tht f is integrle on, nd < m f M for ll in the intervl,. Prove tht m f d M. Use this result to estimte d.. Let f e continuous on the intervl, where f f on,. () Show tht () Use the result in prt () to evlute (c) Use the result in prt () to evlute d......5 5..7..9. 5 5 5 () Produce resonle grph of the velocit function v grphing these points nd connecting them with smooth curve. () Find the open intervls over which the ccelertion is positive. (c) Find the verge ccelertion of the cr (in miles per hour squred) over the intervl,.. (d) Wht does the integrl vt dt signif? Approimte this integrl using the Trpezoidl Rule with five suintervls. (e) Approimte the ccelertion t t.. t f t t dt f v dv dt. f f d f f. lim... n. n n lim 5 5 5... n 5. n n f f f d. sin sin sin d. 7. Verif tht n nn n i i showing the following. () i i i i () n n i i (c) n nn n i i. Prove tht if f is continuous function on closed intervl,, then f d f d. 9. Let I f d where f is shown in the figure. Let Ln nd Rn represent the Riemnn sums using the left-hnd endpoints nd right-hnd endpoints of n suintervls of equl width. (Assume n is even.) Let Tn nd Sn e the corresponding vlues of the Trpezoidl Rule nd Simpson s Rule. () For n n, list Ln, Rn, Tn, nd I in incresing order. () Approimte S.. The sine integrl function sin t Si dt t f i is often used in engineering. The function f t sin t is not t defined t t, ut its limit is s t. So, define f. Then f is continuous everwhere. () Use grphing utilit to grph Si. () At wht vlues of does Si hve reltive mim? (c) Find the coordintes of the first inflection point where >. (d) Decide whether Si hs n horizontl smptotes. If so, identif ech.