# 4.6 - 4.6 Other Integratien Strategies 27 SECTION 4.6...

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Unformatted text preview: 4.6 Other Integratien Strategies 27'? SECTION 4.6 EXERCISES Review nestiens e1 3 at Q. _ , ss./mn,xex 36./—E a 1. Give seme examples ef analytteai metheds fer evaluating ,x; 4, Li + En integrals. in a sin”1 in a: tit 31 fLmLmLJd, 33_ /___m_ e V 1 + 4e 39-46. Geernetry preblerne Use a tabte af int'egrats ta setve the 3. Why might an integral feund in a table differ frem the same inte— fﬂggwmg pmbkmg gral evaluated by a eemputer algebra s y stem? 2. Dees a eemputer algebra system. give an exact result fer an indefi- nite integral? Eenlain. 39. Find the length ef the eurve )1 : erg/'4 en the interval [0, 8]. 4. Is a reduntien fermnia an analytical methed er a numerical methed‘t Explain. 40. Find the length ef the curve 39 == sit? + 3 en the interval [0, 2]. Basie Skills 41. Find the length ef the curve 39 = ei‘ en the interval [0, in 2], 5-22. Table leeknp integrals Use a tabte ef integrate ta determine the fettewing indefinite integrate. 42. The regien beunded by the graph ef y = n2 in a: and the e-aeis en the intervai [1, e] is revelved abeut the .r-aeis. What is the vei- urne ef the selid that is termed? S. / eee‘1 s at 6. / sin 3n ees 22: its 3 b h f 1 4 . The re ien ennded b the ra e e _. and the e-asis 7 rte 3 ' «it: en the interval [0, 12] is reveived abeut the y-aeis. What is the ' m ' m veleme ef the selid that is termed? 3a g}, 44. Find the area ef the regien beunded by the graph ef 9 / an 11}. /~— 1 2” + 7 Jim)" + 9) y = mend the needs betweene: = 0 ands = 3. Veg — at + 2 ctr tit 11' 1 __ ees 4s: 12' - em 45. The regien beunded by the graphs ef y = 17/2, 3: = sin—1.1:, and the git-axis is revelved abent the jut-axis. What is the velame ef the seh'd that is in ed?Ir 13. /——I—~ it): 14. fix/4t +' 12 at “n V4.1" ‘l‘ i 2 7 46. The graphs ef f(:t') = and g[e) : —-——-—— are shevvn an 10 91;; x2 + 1 4 e2 + 1 15. “CE—2;— t I 3“ 3“ 15- _— in the figure. Which is greater, the average value ef f er that ef g I — “30 en the interval [—1, 1]? 21. flnzstttx 22. fnﬂeﬁxar 23—33. Preliminary ever]; Use a tattte af integrate ta rteterrnine the fattening indeﬁnite integrate. These integrate require pretintinar}: were, seen as eentpteting the eaaare er changing variabtee, hefere they _ can b“; fenntt tn a tattte. B 47-54. Indeﬁnite integrals Use a earnpater at gebra eyetent ta evata— ate the fettewing indeﬁnite integrate. Asearne that a is a pesitive reat 23. an2+lﬂxttx.I}U 24' /eg2—8xrtr,e3r8 number. it “mm“? 41/wa 4s. /V42+36dr 25. f—mw—H—w 26.] a2—4x+8¢tx max _ x s3 + 2x + 1e 27‘ f+ 23_ /_3_§Lm 49. fiat}: Fertile 5i}. V/‘(ttg -' t2)_2 tit eh: +1) t(t — 256) a __ 2 an ex ctr (r e ) 52 as 29 ,x?6 30 /—,.r:r(] 51/ ﬁx a" as I2 ‘_ 64:: V32 + 103;; I ICE??? ,1: ) #43 34. ———“"ctx‘ n2s+ Zsina Vi: / ces e ens? 1' W si 278 CHAPTER 4 - lnrnanarren TECHNIQUES 55—62. Definite integrals Use a camputer algebra system ta evaluate the felicwing deﬁnite integrals. In each case. ﬁnci an exact value cf the integral (ahtaineti lay a synthalic methad) antifinti an apprasintate value (ahtaineti by a numerical methad). Carneare the results. 4th an 55. f a;3 iii: 56. / cus'5 a: its: 233 e 4 I - ..1 57. / (9+it)3l2ax 58. / 31“ “tax it LIE +3": “HIE till Ear ﬁll 59. ~————-—§- as. mum—i ﬂ 1+tant g (4+2s1nt} l tar/4 61. /(1ns)ln(1+ .r) tit 62. f in (1 + tanu)tix n e Further Eapleratiens 63. Explain why er why net Determine whether the fellewing state ments are true and give an explanatien er ceuntereaample. a. It is pessibie that a cemputer algebra system says g); . f— w in (a: - 1) — ins and stable at 1ntegrais .r(.'t: — l] sas/ dx i ——-——-—-—=n y .r(.'r-1) h. A ccmputer algetlrra s ystem werking in synthetic made ceuld give the result ﬂag (it = i: and a cempnter algebra system werking in apprnaimate (numerical) metie ceuld give the result Sign = (111111111. a—l I he 64. Apparent discrepancy Three different cumputer algebra systems give the feilnwing results: tit: fﬁn—“ECHS a: \$511203" .1; —Etﬂﬂ Esplain hew they can all be current. 65. Recenciling results Using ens cemputer algebra system. . air sins: - 1 . It was fennel that f—T— = W. and usmg anether l + slur ens .t . its cemputer algebra system. 1t was feunti that m-:—- E l + sins: 2 sin (at/2) u__.._..._____ Recenciie the twe ans s. ces (Jr/2) + sin (#2) wer 66. Apparent discrepancy Reselve the apparent discrepancy between a r mmiﬁn+21 W : —.1 ...___..__._ + C a /u(.r — l)(.r + 2) 6 ﬂ lulﬂ an / a: _umu~1\+mn+2| unl+c s(s *1)[.t + 2) _ 3 f: 2 ' 67—70. Reductinn fnrmulas Use the reductian farrnulas in a table af integrals ta evaluate the feilawing integrals. 67. fit {their 58. fat e"3P as 69. ftan4 3y ti}? 70. fsec‘i 4l'dl 77. 3 71—74. Dnuhle table lenkup The fellawing integrals may require mare than see table laahup. Evaluate the integrals using a table af integrals; then check yaur answer with a carnputer algebra system. 71. fat sin—l 2x615: t _l 73. females I 72. fax ens—1 lﬁsrtt -—1 . 74. f” fleas 2:: e I 75. Evaluating an integral witheut the Fundamental Thenrern {if Calculus Evaluate fﬂwﬂln (l + tan .r) air using the fellewing steps. a. If f is integrable en [0. it]. use substitutiun te shew that a an / ﬁx) a = f are wu- — a) u U! [l tana + tanB I). Use part (a) and the identity tan(a: + E) 2 WE — anus an tn evaluate “ﬁllhﬁl + tan a) tit. (Saurce: The Callege Mathematics Jaurnal 33. 4. Sep 2004) 76. Twe integratinn appruaches Evaluate f ces (ln s) cit twe differ-u ent ways: it. Use tables after first using the substitutien u =- in .x. I). Use integratien by parts twice te verify yeur answer has part (a). Applicatiens Perind [if a pendulum Censider a pendulum at length L meters swinging enlv under the inﬂuence at gravity. Suppuse the pendu— lum starts swinging with an initial displacement hf a} radians (see ﬁgure). The perind (time te cemplete ene full cycle) is given by all 4 a’ T 2 _f _....__t'__..__! i” e "Vi - hgsingal where m3 I g/L. g w 9.3 mls2 is the acceleratien due te grav- ity. and k2 = sin2 (9.3/2). Assume L = 9.8 m. which means at = 1 s_l. a. Use a camputer algebra system ta find the perietl ef the pendu~ lum fer 6.3, 2 ll]. [1.2, . . . . 0.9. 1.0 rad. b. For small. values at an, the parted sheulti be appreaimately 2n“ secends. Far what values cf ﬁg are yeur cemputerl values within 10% cf 2a- (relative errer less than 0.1)? 4.7 Numerical tntegratien 27") Additional Exercises a. Use a cemputer algebra system in eenfi rm this result fer E 73. Are length cf a parabela Let Me) be the length at the parahela n _E 2* '3’ 4* and 5' _ # ﬁx) 3 Is frem a: = 0 te s: = a, where L, a l) is a censtant. h. Evaluate the‘mtegrals with n _. 10 and cenfirm the result. e. Using graphing and / er synthetic cemputatt en, determine whether the values ef the integrals increase er decrease as it increases. it. Find an expressien fer L and graph the functieu. b. Is L cencave up er cencave dewn en [t], w )‘l c. Shaw that as e beenmes large and pesitive, the arc length fune- . . 1 . m 2 . "HIE til: hen increases as a _. that 1s, L(c) kc , where it is a censtant. 34- A remarkable integral It is a fact that f 1 + t m = E fer 79—»82. Deriving fermulas Evaluate the failawing integrals. Assume a all real numbers at. - ll an .t: and s are real numbers and n is an integer. 3. Graph the integrand fer m m _...2! _3/25 -1! _1 f2! 0, 1 ﬁt a: 1, 3 /2, and 2, and explain geemeuicallv hew the area under 79- far 4, l: ill? (USE ll 3 ill? “l‘ 5-) the curve en the interval [0. “Fl/2] remains censtant as in varies. x 11. Use a cent star at ebra s stem is eent‘irm that the inte rat is an. M—eai (Use a2 = as + a) p g y g W censtant fer all re. 31. fate + sear (Useu = as + a) 1 QUICK CHECK ANSWERS 32' ff Sm “ix (Ugemmgmmn by pms') 1. 1 2. Because sings: = 1 — this2 it, the twe results 33. Pewers ef sine and ensine It can be shewn that differ by a censtant, which can be abserbed in the arbitrary n12 WE censtant C. 3. The secend result agrees with the first fer / sin" Mix : mgr: mg; 2 .r :2? 0 after using In a - ln .5 w in (a/b]. The secentl result n e sheuld have abselute values and an arbitrary ceustant. «It 1'3-5“ (n"1)vr ,1, 3,2. 't 2‘4‘6 __H 2 in.— 1saueven1neger 2 a - e — - (n — 1) , , if n E 3 is an edti integer. 3 - 5 - 7 * - n 4.7 Numerical Integration _ Situatieus arise in which the analytical metheds we have develeped se far cannet be used te evaluate a definite integral. Fer example, an integrand may net have an ebvieus amides rivative (such as sea 3:2 and l / In 3:), er perhaps the value ef the integrand is knewn enly at a finite set ef paints, which makes finding an antiderivative impessihle. When analytical metheds fail, we eften turn te numerical urethane, which are typiw cally dune en a calculater er cemputer. These metheds tie net preduce exact values ef definite integrals, but theyr previde appreximatiens that are generally quite accurate. Many calculaters, seftware packages, and cemputer algebra systems have built-in numerical in- tegratieu metheds. In this sectien, we explere seme ef these methetls. I-..-\_-\....._n.--\..-..-....| ..|-|..-|-. .|.|.a-.-\. ..-..-\_-\.....-|.- .-\. Abselute and Relative Errer Because numerical metheds (in net typically preduce exact results, we sheulti be can“ cernecl sheet the accuracy'ef appreaimatiens, which leads in the ideas at assaults and relative erra r. ___. -_.-. -_. __ _ .L 1... .. _._... ___ _ ..... ___—5-5., . ___ ___ “_._. _ .,. . -\..---- _._“... ..,_..u . _._ , -_.._.___-_. _._,__ __,,__._...w_,,. u... _ _.-..1..._-_ _.. . ___wu _._“... _ _ _._ I ___—_._.-.__- . _....._..,..._..._. . --\.--\.- ..._..,._._ ._. .._-- . ._.,.,_.,I _.....r... 5.-.. u... |--- “huh.”— ..,__...._,__,..,. .ﬂw. ewes Ahsulute and Relative Errer Suppese e is a cemputecl numerical salutien te a preblem having an exact selutien it. There are twe cemrnen measures ef the errer in e as an appresimatien te .r: absnlute errer = ls - .rl i and 1wa 1|" Because the erect salutien is usually net relative EI‘I‘ﬂr : knewn, the geal in practice is te estimate ' _ l}: l i the maximum size ef the errer. I ...
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