1 6_88.qd // : PM Pge CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls Section 8.8 f() = d The unounded region hs n re of. Figure 8.7 Improper Integrls Evlute n improper integrl tht hs n infinite limit of integrtion. Evlute n improper integrl tht hs n infinite discontinuit. Improper Integrls with Infinite Limits of Integrtion The definition of definite integrl f d requires tht the intervl, e finite. Furthermore, the Fundmentl Theorem of Clculus, which ou hve een evluting definite integrls, requires tht f e continuous on,. In this section ou will stud procedure for evluting integrls tht do not stisf these requirements usull ecuse either one or oth of the limits of integrtion re infinite, or f hs finite numer of infinite discontinuities in the intervl,. Integrls tht possess either propert re improper integrls. Note tht function f is sid to hve n infinite discontinuit t c if, from the right or left, lim f c or To get n ide of how to evlute n improper integrl, consider the integrl d which cn e interpreted s the re of the shded region shown in Figure 8.7. Tking the limit s produces d lim lim f. c d lim. This improper integrl cn e interpreted s the re of the unounded region etween the grph of f nd the -is (to the right of ). Definition of Improper Integrls with Infinite Integrtion Limits. If f is continuous on the intervl,, then f d lim f d.. If f is continuous on the intervl,, then f d lim f d.. If f is continuous on the intervl,, then c f d f d f d c where c is n rel numer (see Eercise ). In the first two cses, the improper integrl converges if the limit eists otherwise, the improper integrl diverges. In the third cse, the improper integrl on the left diverges if either of the improper integrls on the right diverges.
2 6_88.qd // : PM Pge 579 SECTION 8.8 Improper Integrls 579 EXAMPLE An Improper Integrl Tht Diverges Diverges (infinite re) = This unounded region hs n infinite re. Figure 8.8 Evlute Solution d d. lim See Figure 8.8. d lim ln lim ln Tke limit s. Appl Log Rule. Appl Fundmentl Theorem of Clculus. Evlute limit. NOTE Tr compring the regions shown in Figures 8.7 nd 8.8. The look similr, et the region in Figure 8.7 hs finite re of nd the region in Figure 8.8 hs n infinite re. EXAMPLE Improper Integrls Tht Converge Evlute ech improper integrl.. e d. d Solution. e d lim e d. d lim d See Figure 8.9. lim e lim e See Figure 8.. lim rctn lim rctn = e = + The re of the unounded region is. Figure 8.9 The re of the unounded region is. Figure 8.
3 6_88.qd // : PM Pge CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls In the following emple, note how L Hôpitl s Rule cn e used to evlute n improper integrl. EXAMPLE Using L Hôpitl s Rule with n Improper Integrl = ( )e The re of the unounded region is e. Figure 8. Evlute Solution Use integrtion prts, with dv e d nd u. Now, ppl the definition of n improper integrl. Finll, using L Hôpitl s Rule on the right-hnd limit produces lim from which ou cn conclude tht See Figure 8.. e d. e d e e d e lim e e d e. e e e C e C e d lim e lim e e EXAMPLE Infinite Upper nd Lower Limits of Integrtion Evlute Solution Note tht the integrnd is continuous on,. To evlute the integrl, ou cn rek it into two prts, choosing c s convenient vlue. e e d e e d e e d e e d. e = + e The re of the unounded region is Figure 8.. See Figure 8.. lim rctn e lim rctn e lim rctn e lim rctn e
4 6_88.qd // : PM Pge 58 SECTION 8.8 Improper Integrls 58 EXAMPLE 5 Sending Spce Module into Orit In Emple of Section 7.5, ou found tht it would require, mile-tons of work to propel 5-metric-ton spce module to height of 8 miles ove Erth. How much work is required to propel the module n unlimited distnce w from Erth s surfce? Solution At first ou might think tht n infinite mount of work would e required. But if this were the cse, it would e impossile to send rockets into outer spce. Becuse this hs een done, the work required must e finite. You cn determine the work in the following mnner. Using the integrl of Emple, Section 7.5, replce the upper ound of 8 miles nd write W,, d The work required to move spce module n unlimited distnce w from Erth is pproimtel 6.98 foot-pounds. Figure 8. lim,, lim,,,, 6, mile-tons 6.98 foot-pounds. See Figure 8.. Improper Integrls with Infinite Discontinuities The second sic tpe of improper integrl is one tht hs n infinite discontinuit t or etween the limits of integrtion. Definition of Improper Integrls with Infinite Discontinuities. If f is continuous on the intervl, nd hs n infinite discontinuit t, then c f d lim f d. c. If f is continuous on the intervl, nd hs n infinite discontinuit t, then f d lim f d. c c. If f is continuous on the intervl,, ecept for some c in, t which f hs n infinite discontinuit, then c f d f d f d. c In the first two cses, the improper integrl converges if the limit eists otherwise, the improper integrl diverges. In the third cse, the improper integrl on the left diverges if either of the improper integrls on the right diverges.
5 6_88.qd // : PM Pge CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls = (, ) Infinite discontinuit t Figure 8. EXAMPLE 6 Evlute d. An Improper Integrl with n Infinite Discontinuit Solution The integrnd hs n infinite discontinuit t, s shown in Figure 8.. You cn evlute this integrl s shown elow. d lim lim EXAMPLE 7 An Improper Integrl Tht Diverges d Evlute. Solution Becuse the integrnd hs n infinite discontinuit t, ou cn write d lim lim. 8 So, ou cn conclude tht the improper integrl diverges. EXAMPLE 8 An Improper Integrl with n Interior Discontinuit d Evlute. = Solution This integrl is improper ecuse the integrnd hs n infinite discontinuit t the interior point, s shown in Figure 8.5. So, ou cn write d d d. From Emple 7 ou know tht the second integrl diverges. So, the originl improper integrl lso diverges. The improper integrl Figure 8.5 d diverges. NOTE Rememer to check for infinite discontinuities t interior points s well s endpoints when determining whether n integrl is improper. For instnce, if ou hd not recognized tht the integrl in Emple 8 ws improper, ou would hve otined the incorrect result d 8 8. Incorrect evlution
6 6_88.qd // : PM Pge 58 SECTION 8.8 Improper Integrls 58 The integrl in the net emple is improper for two resons. One limit of integrtion is infinite, nd the integrnd hs n infinite discontinuit t the outer limit of integrtion. EXAMPLE 9 A Doul Improper Integrl = ( + ) The re of the unounded region is. Figure 8.6 Evlute Solution To evlute this integrl, split it t convenient point (s, ) nd write d d d lim rctn lim c rctn c See Figure 8.6. d.. EXAMPLE An Appliction Involving Arc Length Use the formul for rc length to show tht the circumference of the circle is. =, The circumference of the circle is. Figure 8.7 Solution To simplif the work, consider the qurter circle given, where. The function is differentile for n in this intervl ecept. Therefore, the rc length of the qurter circle is given the improper integrl s This integrl is improper ecuse it hs n infinite discontinuit t. So, ou cn write s lim. d d. d rcsin d Finll, multipling, ou cn conclude tht the circumference of the circle is s, s shown in Figure 8.7.
7 6_88.qd // : PM Pge CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls This section concludes with useful theorem descriing the convergence or divergence of common tpe of improper integrl. The proof of this theorem is left s n eercise (see Eercise 9). THEOREM 8.5 d p, if p > p diverges, if p A Specil Tpe of Improper Integrl EXAMPLE An Appliction Involving A Solid of Revolution FOR FURTHER INFORMATION For further investigtion of solids tht hve finite volumes nd infinite surfce res, see the rticle Supersolids: Solids Hving Finite Volume nd Infinite Surfces Willim P. Love in Mthemtics Techer. To view this rticle, go to the wesite The solid formed revolving (out the -is) the unounded region ling etween the grph of f nd the -is is clled Griel s Horn. (See Figure 8.8.) Show tht this solid hs finite volume nd n infinite surfce re. Solution to e Using the disk method nd Theorem 8.5, ou cn determine the volume The surfce re is given Becuse V d. Theorem 8.5, p > S f f d d. > on the intervl,, nd the improper integrl diverges, ou cn conclude tht the improper integrl d d lso diverges. (See Eercise 5.) So, the surfce re is infinite. f() =, FOR FURTHER INFORMATION To lern out nother function tht hs finite volume nd n infinite surfce re, see the rticle Griel s Wedding Cke Julin F. Fleron in The College Mthemtics Journl. To view this rticle, go to the wesite Griel s Horn hs finite volume nd n infinite surfce re. Figure
8 6_88.qd // : PM Pge 585 SECTION 8.8 Improper Integrls 585 Eercises for Section 8.8 In Eercises, decide whether the integrl is improper. Eplin our resoning. d d ln d 5 6 d In Eercises 5, eplin wh the integrl is improper nd determine whether it diverges or converges. Evlute the integrl if it converges d d 9. e d. Writing In Eercises, eplin wh the evlution of the integrl is incorrect. Use the integrtion cpilities of grphing utilit to ttempt to evlute the integrl. Determine whether the utilit gives the correct nswer... d 5 e d d 5 d d 8 9. e d. In Eercises 5, determine whether the improper integrl diverges or converges. Evlute the integrl if it converges d d d d 9.. e d e d.. e d e d.. e cos d e sin d, > ln d ln d e d d e d e. e d. cos d sin d In Eercises 8, determine whether the improper integrl diverges or converges. Evlute the integrl if it converges, nd check our results with the results otined using the integrtion cpilities of grphing utilit... d d 7. ln d tn d.... d. d d 8 d d ln d See for worked-out solutions to odd-numered eercises d e sec d 8 d ln d sec d d d
9 6_88.qd // : PM Pge CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls In Eercises 9 nd 5, determine ll vlues of p for which the improper integrl converges d d p p 5. Use mthemticl induction to verif tht the following integrl converges for n positive integer n. n e d 5. Given continuous functions f nd g such tht f g on the intervl,, prove the following. () If converges, then g d f d converges. () If diverges, then f d g d diverges. In Eercises 5 6, use the results of Eercises 9 5 to determine whether the improper integrl converges or diverges d d e d d d d 59. d 6. d 6. e d 6. ln d Writing Aout Concepts 6. Descrie the different tpes of improper integrls. 6. Define the terms converges nd diverges when working with improper integrls. 65. Eplin wh 66. Consider the integrl d. d. To determine the convergence or divergence of the integrl, how mn improper integrls must e nlzed? Wht must e true of ech of these integrls if the given integrl converges? Are In Eercises 67 7, find the re of the unounded shded region. 67. e, < 68. ln 69. Witch of Agnesi: 7. Witch of Agnesi: Are nd Volume In Eercises 7 nd 7, consider the region stisfing the inequlities. () Find the re of the region. () Find the volume of the solid generted revolving the region out the -is. (c) Find the volume of the solid generted revolving the region out the -is. 7., 7. e,, 7. Arc Length Sketch the grph of the hpoccloid of four cusps nd find its perimeter. 7. Arc Length Find the rc length of the grph of 6 over the intervl,. 75. Surfce Are The region ounded 8 6 6, is revolved out the -is to form torus. Find the surfce re of the torus. 76. Surfce Are Find the re of the surfce formed revolving the grph of e on the intervl, out the -is. 6 6
10 6_88.qd // : PM Pge 587 SECTION 8.8 Improper Integrls 587 Propulsion In Eercises 77 nd 78, use the weight of the rocket to nswer ech question. (Use miles s the rdius of Erth nd do not consider the effect of ir resistnce.) () How much work is required to propel the rocket n unlimited distnce w from Erth s surfce? () How fr hs the rocket trveled when hlf the totl work hs occurred? ton rocket 78. -ton rocket Proilit A nonnegtive function f is clled proilit densit function if f t dt. The proilit tht lies etween nd is given P f t dt. The epected vlue of is given E tft dt. In Eercises 79 nd 8, () show tht the nonnegtive function is proilit densit function, () find P, nd (c) find E ft 7 et7,, ft 5e t5,, Cpitlized Cost In Eercises 8 nd 8, find the cpitlized cost C of n sset () for n 5 ers, () for n ers, nd (c) forever. The cpitlized cost is given n C C cte rt dt where C is the originl investment, t is the time in ers, r is the nnul interest rte compounded continuousl, nd ct is the nnul cost of mintennce. 8. C $65, 8. C $65, ct $5, r.6 where N, I, r, k, t t < t t < nd c re constnts. Find P. ct $5,.8t r.6 8. Electromgnetic Theor The mgnetic potentil P t point on the is of circulr coil is given P NIr k d c r 8. Grvittionl Force A semi-infinite uniform rod occupies the nonnegtive -is. The rod hs liner densit which mens tht segment of length d hs mss of A prticle of mss m is locted t the point,. The grvittionl force F tht the rod eerts on the mss is given F where G is the grvittionl constnt. Find F. True or Flse? In Eercises 85 88, determine whether the sttement is true or flse. If it is flse, eplin wh or give n emple tht shows it is flse. 85. If f is continuous on, nd lim f, then f d converges. 86. If f is continuous on, nd f d diverges, then f. lim 87. If is continuous on, nd lim f, then f f d f. 88. If the grph of f is smmetric with respect to the origin or the - is, then converges if nd onl if f d f d converges. 89. Writing () The improper integrls nd d diverge nd converge, respectivel. Descrie the essentil differences etween the integrnds tht cuse one integrl to converge nd the other to diverge. () Sketch grph of the function sin over the intervl,. Use our knowledge of the definite integrl to mke n inference s to whether or not the integrl sin d converges. Give resons for our nswer. (c) Use one itertion of integrtion prts on the integrl in prt () to determine its divergence or convergence. 9. Eplortion Consider the integrl GM d tn n d where n is positive integer. d () Is the integrl improper? Eplin. () Use grphing utilit to grph the integrnd for n,, 8, nd. (c) Use the grphs to pproimte the integrl s n. (d) Use computer lger sstem to evlute the integrl for the vlues of n in prt (). Mke conjecture out the vlue of the integrl for n positive integer n. Compre our results with our nswer in prt (c). d.
11 6_88.qd // : PM Pge CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls 9. The Gmm Function The Gmm Function n is defined n n e d, () Find,, nd. () Use integrtion prts to show tht n nn. (c) Write n using fctoril nottion where n is positive integer. 9. Prove tht I n n where n I n, I n n d, n. n Then evlute ech integrl. () () d (c) d 5 5 d 6 Lplce Trnsforms Let f t e function defined for ll positive vlues of t. The Lplce Trnsform of f t is defined Fs e st ft dt if the improper integrl eists. Lplce Trnsforms re used to solve differentil equtions. In Eercises 9, find the Lplce Trnsform of the function. 9. ft 9. ft t 95. ft t 96. ft e t 97. ft cos t 98. ft sin t 99. ft cosh t. ft sinh t. Norml Proilit The men height of Americn men etween 8 nd ers old is 7 inches, nd the stndrd devition is inches. An 8- to -er-old mn is chosen t rndom from the popultion. The proilit tht he is 6 feet tll or tller is P7 < 7 n >. (Source: Ntionl Center for Helth Sttistics) e 7 8 d. () Use grphing utilit to grph the integrnd. Use the grphing utilit to convince ourself tht the re etween the -is nd the integrnd is. () Use grphing utilit to pproimte P7 <. (c) Approimte.5 P7 7 using grphing utilit. Use the grph in prt () to eplin wh this result is the sme s the nswer in prt ().. () Sketch the semicircle. without evluting either integrl.. For wht vlue of c does the integrl c d converge? Evlute the integrl for this vlue of c.. For wht vlue of c does the integrl c d converge? Evlute the integrl for this vlue of c. 5. Volume Find the volume of the solid generted revolving the region ounded the grph of f out the -is. 6. Volume Find the volume of the solid generted revolving the unounded region ling etween ln nd the -is out the -is. u-sustitution In Eercises 7 nd 8, rewrite the improper integrl s proper integrl using the given u-sustitution. Then use the Trpezoidl Rule with n 5 to pproimte the integrl () Eplin wh d d f ln,, sin d, cos d, < u u 9. () Use grphing utilit to grph the function e () Show tht e d. ln d.. Let f d e convergent nd let nd e rel numers where. Show tht f d f d f d f d.