1 _R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite In Eercises 8, find the indefinite integrl 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, go to the wesite Use integrtion to find the prticulr solution of the differentil eqution nd use grphing utilit to grph the solution..,,.,, f sin 5 cos sec 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 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 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 positive integers, nd (c) 8... n.. Evlute ech sum for,, 5,, nd 5 7. See for worked-out solutions to odd-numered eercises. 5 5 i i 5 i (c) (d) 5 i i i i... n n i i ii i 5 i i i 7
2 _R.qd // : PM Pge 7 REVIEW EXERCISES 7 In Eercises 7 nd 8, use upper nd lower sums to pproimte the re of the region using the indicted numer of suintervls of equl width In Eercises, use the it 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 it 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 it s definite integrl on the intervl [, ], where is n point in the ith suintervl Limit n i n i In Eercises 7 nd 8, set up definite integrl tht ields the re of the region. (Do not evlute the integrl.) ,,.,,. 5,,.,, c i i c i c i c i i 7. f 8. f 8 Intervl,, In Eercises nd, sketch the region whose re is given the definite integrl. Then use geometric formul to evlute the integrl (c) f g. (d) 5f.. Given f nd f, evlute (c) In Eercises 5, use the Fundmentl Theorem of Clculus to evlute the definite integrl. (d).. 5. t t dt sin d 5. In Eercises 5 5, sketch the grph of the region whose re is given the integrl, nd find the re In Eercises 57 nd 58, determine the re of the given region. 57. sin 58. cos 5. Given f nd g, evlute f g. f. f. π f. f g. f. t dt 5 sec t dt π π π
3 _R.qd // : PM Pge 8 8 CHAPTER Integrtion In Eercises 5 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. 5..,,, sec,,, 8.,., 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,., 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 7. sin cos 7. sin sin cos cos sin 77. tn n sec, n 78. sec tn 7. sec sec 8. In Eercises 8 88, evlute the definite integrl. Use grphing utilit to verif our result cos 88. In Eercises 7 8, find the indefinite integrl d tn cot csc d 8 sin Slope Fields In Eercises 8 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, go to the wesite In Eercises nd, find the re of the region. Use grphing utilit to verif our result 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. 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.. 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 5 8, 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 cos 8. sin t t p dt. cos sin π π
4 _R.qd // : PM Pge P.S. Prolem Solving P.S. Prolem Solving See for worked-out solutions to odd-numered eercises.. Let L >. t dt, 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 G. (c) Use the definition of the derivtive to find the ect vlue of the it G. In Eercises nd, write the re under the grph of the given function defined on the given intervl s it. Then use computer lger sstem to evlute the sum in prt, nd (c) evlute the it 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 , 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,.. The Two-Point Gussin Qudrture Approimtion for f is f f f. Use this formul to pproimte cos. Find the error of the pproimtion. Use this formul to pproimte. (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 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. 8. Glileo Glilei (5 ) stted the following proposition concerning flling ojects: The time in which n spce is trversed uniforml ccelerting o is equl to the time in which tht sme spce would e trversed the sme o moving t uniform speed whose vlue is the men of the highest speed of the ccelerting o nd the speed just efore ccelertion egn. Use the techniques of this chpter to verif this proposition.. The grph of the function f consists of the three line segments joining the points,,,,,, nd 8,. The function F is defined the integrl F f t dt. Sketch the grph of f. Complete the tle. F h (c) Find the etrem of F on the intervl, 8. (d) Determine ll points of inflection of F on the intervl, 8.
5 _R.qd // : PM Pge 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 it. Use n pproprite Riemnn sum to evlute the it 5. Suppose tht f is integrle on, nd < m f M for ll in the intervl,. Prove tht m f M. Use this result to estimte.. 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 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.8. t f t t dt f v dv dt. f f f f.... n. n n n 5. n n f f f. sin sin sin. 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 8. Prove tht if f is continuous function on closed intervl,, then f f.. Let I f 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 it 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.