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Section 6.1 7th Ed - 394 CHAPTER 6 Techniques of...

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Unformatted text preview: 394 CHAPTER 6 Techniques of Integration . ' x P R E R E Q U I 5] T E The following warm-up exercises involve skills that were covered in earlier sections. You will R E V | E w 6 , 1 use these skillsin the exercise set for this section. l“ ln Exercises l—8, evaluate the indefinite integral. 1. J- 5 dx 2. %dx 5. J2x(x2 + 1)3 dx 6. J3x2(x3 — Uzdx 7 666‘ dx 8 2 dx ' ' 2x + 1 ‘ ln Exercises 9—12, simplify the expression. 9. 2x(x — 1)2 + x(x — 1) 10. 6x(x + 4)3 — 3x2(x + 4)2 5 11. 3(x + 7)”2 — 2x(x + 7)-1/2 12. (x + 5)”3 — 5(x + ski/3 l .___ J g i l ln Exercises 1—38,find the indefinite integral. an e; 25. _ [ix 26. e X + 2 1 + e‘ . 1. ( — 2)4 dx 2. (x + 5)3/2dx 2 x x 2 . . 2 4 7 f(x+1)4dx 28 f(x+1)3dx 3. 7 dt 4. 3 dt (1‘ — 9)“ (1 — I) x 5x 29. —-——7 dx 30. 3 dx 2: — 1 2y3 (3x — 1)" (x — 4) .1 5. 7 d 6. 4 db} Elil.‘ t"—t+2 y+1 1 1 i ;. 31. \fl—ldz 32. f+1dx l 7.J'\/1+xdx 8.J(3+x)mdx 2./ x f , 1 + + ' 12 +2 “+2 33.J——-—tt 1dr 34.J——6xx xdx 9. ’2‘ [ix 10. x3 dx 3x + x x + x x x2 ‘ 1 1 35. —2‘x‘+—1 dx 36. 1 dx ! ‘9 11. [mdx 12. de x R} 1 1 37. [£241 * Id! 38. JyZE/y + ldy 13. dx l4. ————— (ix 1 J\/x+1 \/5x+1 . “ 3.1 4 2‘- In Exerctses 39—46, evaluate the definite integral. 15. e “an 16. J ’3 dx 4 4 1—6' Ha" 39. ./2x+1dx 4o. ./4x+1.11 o z 1 17. i. 18. , 2 \Vx+1 41. 3xex'dx 42. f e”1"dx O (x + 5 21. x x2 + 4 dx 22. (it 0. 0.5 \/1_ 1‘2 45. I x(1— x)3dx 46. I x2(1 — x)3dx O O 23. e3" d): 24. x2 2x 0 19. J .11- 20. I dx 4 x 1 A—1 A~4 43. de 44.fx(x+5)4dx l l ° 4) 0 Will ix - 5:)3 dx SECTION 6.1 integration by Substitution 395 In Exercises 47—54, find the area of the region bounded by the 63. Probability The probability of recall in an experiment is graphs of the equations.Then use a graphing utility to graph the modeled by region and verify your answer. 1, 15 _ 47.y=x\/x_—§,y=0,x=7 P(aSXSb):aTx‘/1_de "5 43. y=x\/2.x+ l, y=0, x=4 where x is the percent of recall (see figure). (a) What is the probability of recalling between 40% and 50_ y = xzm, y = 0, x = 7 80%? x2 _ 1 (b) What is the median percent recall? That is, for what 5Ly=mmiy=0ix=lfl= valueofbisP(OSbe)=O.5? _ 2x *1 _ _ l . _ v 15 y 52.y—— x+3,y-O,x—2,A—6 A y=—4—x\/l——_x y:331%x3(1~x)3/2 V 3__ _ _ 3 ~ — _ __ 53'y_x X+1’y"0’x‘0’x_ 2P(a5x5b) P(aEbe) 54. y=x\3/x—2, y=0,x=2,x=10 7" In Exercises 55—58, find the area of the region bounded by the 1" graphs of the equations. ‘1 bos 1 I 55.y=~x\/x+2,y=0 56.y=x\3/1—x,y=0 050 b v Figure for 63 Figure for 64 64. Probability The probability of finding between a and b 05 percent iron in ore samples is modeled by b P(a S x S b) =J 1155x3(1— x)3/2 dx _ _\. a 32 _2 _1 0'5 1'0 (see figure). Find the probabilities that a sample will con— 57_ y2 = x2(1 __ x2) 58. y 2 MO + fl), tain betwien (a) 0% and 25% and (b) 50% and 100% iron. ' 65. t D ' t — ' ‘ ' (Hint: Find the area of y : 01 x = O, x = 4 Me com ogy unng a we week period in March m'a , small town near Lake Erle, the measurable snowfall 5 (1n the region bounded by 3’ inches) on the ground can be modeled by S(t)=t 14—t, Osrs 14 where t represents the day. y=x\/1—x2andy= 0. Then multiply by 4.) (21) Use a graphing utility to graph the function. (b) Find the average amount of snow on the ground during the two-week period. (c) Find the total snowfall over the two-week period. l . . ‘ 66. Revenue A company sells a seasonal product that gen— n Exerctses 59 and 60, find the volume ofthe solid generated by erates a daily revenue R (in dollars per year) modeled by reVOlving the region bounded by the graphis) of the equation(s) about the x—axis. R = 0.06:2(365 * t)”2 + 1250, O S r S 365 59. y = x /1 _ x2 where 2‘ represents the day. 50. y : VG“ _ x)2, y 2 O (a) Find the average daily revenue over a period of 1 year. (b) Describe a product whose seasonal sales pattern resem- In Exercises 61 and 62, find the average amount by which the bles the model. Explain your reasoning. fUnction fexceeds the function g on the interval. 1 x In Exercises 67 and 68, use a program similar to the Midpoint Rule 61- fix) = x + 1’ 8(1‘) : (x + 07’ [0, 1] program on page 366 with n = 10 to approximate the area of the region bounded by the graph(s) of the equation(s). 62- f(x) = xx/4x + l, g(x) = 2x/F, [0,2 67. y = \3/XJ4 —x, y = 0 68. 3,2 = x2(1 ~ X3) 1 :3 Answers to Selected Exercises A71 ; 1 ‘1 1 1 1 33. 11 = 4: 13.3203 85. 11 = 4: 0.7867 49. Area 2 116.331 1 2 11 = 20: 13.7167 11 = 20; 0.7855 1 1 i 1 87. 111114 z 4.355 89. gel ~ e-Z) -~ 11.394 1 . 5677 271' 577' 1 91. 3 93. 35 95. 16 15 1 SAMPLE POST-GRAD EXAM QUESTIONS . Z (page 386) I 1 1. d 2. b 3. c 4. b 5. a 1 6. b 7. d 8. d 9. a 1 CHAPTER 6 , SECTION 6.1 {page394} 1 55 E/2 57g 59 4——7T~0838 611 1 1 ‘ 15“ ' 3 ' 15 ‘ ' 2 Prerequisite Review 63- (81) 0-547 (13) 0.586 . 5x + C 2. %x + C 3. $15” + C 65' (a) 25 3 . %x5/3 + C 5. 1012 + 1)4 + C 1 . % + C 7. e61 + C 1 o 14 1 ; . lnl2x + 1] ~- C 9. x(x — 1)(2x — 1) 0 . . - 3x(x + 4)2(x + 8) (b) About 13.97 inches (0) About 195.56 inches 1 11. (x + 21)(x ~ 7)-1/2 12. x(x + 5)-2/3 67_ 5.835 ' ‘ 1 1 1 2 SECTION 6.2 11111994031 1 . — — 5 + . — + ‘ f 1 5 (x 2) C 3 9 _ I C 1 5. 111|r2 — 1+ 2| + C 7. §(1 + x)3/2 + C Prerequisite Review 1 . 1 1 1 3 7 7 1 1 2X :1 1 . -— ~ . ————— + 1 - . 9 111(32 + A) + C 11 10(5x + DZ C 1. x + 1 . 3. 3x er 1 13. 2./x + 1 + C 15. —%1n|1 — e3~| + C 5. e-‘(x2 + 2x) 6, emu , 2x) _ 1 17. —%e“3-"3 + C 19. %x2 + x +111|x — 1| + C . _ , 10. s 1 :1 1 21. 1(12 + 4)” + C 23. 1e“ + C 1 3| _1 1 - l 3.\' _ i 3.\' . _ 27. —.v ._ —.\‘ _ —.\' 11 25. —1n|e“ + 2| + C 27. 5+—1)2 + W + C 1 3“ 9g + C 3 x e 2“ 26 + C ‘ 1 (x x ) 5.x1an—x+C 7.1e4-1+C 1 ‘1 29. §<lnl3x —1l— 3x _1>+ C 9. L(4_1:_1)+ C 11: 18“: + C 1 .1 16 2 11 1 31. 2(fr— 1)+21nl\/1~ 1| + C 13. x26“ — 26x + 224' + C 1 33. 4\/1+111|r| + C 35.1(1-1)./2x+1+ C 15-1t21r1|1+1l~11nlt+1|~1(r~1)2 + C 37. {—3750 — 1)3/2[35 — 42(1 — 1) + 15(1 — 02]} + C = / x2 x2 x2 .—“+ .— -7-—— -+—~+ —fi(15:2 + 121+ 8)(1 — r)3/Z + C 17 e C 19 2 (1“ A) 2 I“ 4 C 39. % 41. %(e — 1) z 2.577 21. 1011 x)3 + C 23. fig- — 1)3/2(3x + 2) + C 43.1112 —%= 0.193 45. 342% ...
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