07 - CHAPTER 7 Applications of Integration Section 7.1 Section 7.2 Section 7.3 Section 7.4 Section 7.5 Section 7.6 Section 7.7 Area of a Region Between

07 - CHAPTER 7 Applications of Integration Section 7.1...

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Unformatted text preview: CHAPTER 7 Applications of Integration Section 7.1 Section 7.2 Section 7.3 Section 7.4 Section 7.5 Section 7.6 Section 7.7 Area of a Region Between Two Curves . . . . . . . . . . .2 Volume: The Disk Method . . . . . . . . . . . . . . . . . 18 Volume: The Shell Method . . . . . . . . . . . . . . . . . 33 Arc Length and Surfaces of Revolution . . . . . . . . . . . 44 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Moments, Centers of Mass, and Centroids . . . . . . . . . 61 Fluid Pressure and Fluid Force . . . . . . . . . . . . . . . 73 Review Exercises Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 CHAPTER 7 Applications of Integration Section 7.1 6 Area of a Region Between Two Curves 6 2 1. A 0 0 x2 6x dx 0 x2 6x d x 2. A 2 2 2x x2 2 5 4 dx x2 2x 1 dx 3 1 3. A 0 3 x2 2x 2 0 2x 6x dx 3 x2 4x 3 dx 4. A 0 x2 x3 dx 0 0 1 5. A or 2 1 1 3 x3 x3 0 x dx x dx 6 1 x3 x dx 6. A 2 0 x 1 3 x 1 dx 6 4 7. 0 y 5 4 3 2 1 x 1 x dx 2 1 6 8. 1 1 x2 y 2 x2 1 dx 9. 0 y 6 5 42 x3 x dx 6 −2 x 2 3 2 1 x 1 2 3 4 5 −2 x 1 2 3 4 5 6 3 10. 2 y 7 6 5 4 3 2 1 −1 x3 3 x x dx 3 3 4 11. 3 2 y 3 sec x d x 12. 4 sec2 x y cos x dx (3, 6) 3 π (− 4 , 2) 2 π ( 4 , 2) (2, 2 ) 3 2 (3, 1) x 4 5 6 7 2π 3 π 3 π −1 3 2π 3 x π (− 4 , 22 ) (π , 22 ) 4 −π 4 π 4 x 2 S ection 7.1 Area of a Region Between Two Curves 3 13. (a) x x 4 y2 y y y2 6 2 4 y y 0 0 y2 2 2 −6 −4 6 4 y (0, 2) x 6 3y (− 5, − 3) −6 Intersection points: 0, 2 and 0 5, 4 3 24 0 A 5 2 x 4 3 2 y2 4 y x dx 2 dy x dx 61 6 32 3 125 6 (b) A 125 6 4 9 14. (a) y x 2 x 2 and y 6 6 2 x x 6 0⇒ x 3, 9 3x 2 0 (b) A 0 2 y dy 4 6 125 6 y y dy x ⇒x Intersection points: 2, 4 and y 10 32 3 61 6 (−3, 9) 8 6 4 (2, 4) −6 −4 −2 x −2 2 4 6 2 A 3 6 x x 2 dx 125 6 1 2x 15. f x gx A 4 x x 1 1 2 3 2 y 16. f x (3, 4) 2 2 1 y gx A x 4 3 Matches (d) (0, 1) x 1 2 3 (0, 2) Matches (a) 1 (4, 0) x 1 2 3 2 17. A 0 2 0 13 x 2 13 x 2 x2 2 4 2 2 x 2 x 1 dx 1 dx 6 5 4 3 y (2, 6) x4 8 16 8 (0, 2) 1 −2 (2, 3) x 0 (0, 1) x 1 3 4 2 0 2 4 Chapter 7 8 Applications of Integration 1 x 2 7 x 2 3 xx 8 10 dx 8 18. A 2 8 2 10 32 x 8 7x2 4 112 8 dx 8 6 4 2 y (2, 9) (8, 6) x3 8 64 10x 2 (2, 9 ) 2 (8, 0) x 2 4 6 10 80 1 7 20 18 19. The points of intersection are given by: x2 xx A 0 4 20. The points of intersection are given by: x2 4x x2 1 3x x2 x 0 3x when x 4x 3x dx 6 4x 4 4 0 0 when x gx x2 f x dx 4x d x 4 −1 −2 −3 −4 −5 1 0, 4 y 0, 3 1 dx y (0, 0) 1 2 3 (4, 0) 5 3 x A 0 3 x2 x2 0 1 x 0 x3 3 32 3 2x2 0 x3 3 9 3x2 2 27 2 3 0 5 4 3 2 (3, 4) 9 2 (0, 1) x 1 2 3 4 5 6 21. The points of intersection are given by: x2 x A 1 2 22. The points of intersection are given by: x2 4x x3 3 2x 2x 2 1 1 3x 3 1, 2 A 2 x x 2 0, 3 0 when x f x dx 3 x x2 x2 dx 2x 0 when x g x dx 4x 3x d x 2 x gx 3x 1 2 fx 0 3 1 dx 0 3 x2 x2 y 2 dx y 6 2 1 0 2x x2 2 x3 3 2 1 9 2 10 8 6 (2, 9) x3 3 32 x 2 3 0 9 2 5 4 3 (3, 5) 4 ( 1, 0) x (0, 2) 1 2 −1 1 2 3 4 5 4 3 2 x 23. The points of intersection are given by: x x A 0 y 2 1 1 x and x x 2 y 0 and 2 0 y dy 2y x x y2 0 2 1 3 2 1 0 1 (1, 1) (2, 0) x Note that if we integrate with respect to x, we need two integrals. Also, note that the region is a triangle. (0, 0) 2 3 S ection 7.1 5 5 1 Area of a Region Between Two Curves 5 24. A 1 y 1.0 0.8 1 x2 0 dx 1 x 4 5 25. The points of intersection are given by: 3x 1 3x 3 x 1 0, 3 (1, 1) x when x fx g x dx 1 x dx x A 0.6 0.4 0.2 0 3 3x (5, 0.04) x 1 2 3 4 5 1 dx y 0 3 3x 0 12 5 2 3x 9 32 x2 2 3 0 3 2 4 3 2 (3, 4) (0, 1) −2 x 1 2 3 4 26. The points of intersection are given by: 3 x x 1 1 2x 2 1 x x 0 0 1 1 3 y x3 3x2 3x 1 1 (2, 1) (1, 0) 2 x x3 x x2 xx A 2 3x2 3x 2x 1 0⇒x 1 3 0, 1, 2 1 dx 1 (0, −1) x 0 x 1 2 2 x2 2 1 2 x 1 3 x 4 0 43 0 3 4 1 2 28. The points of intersection are given by: 2y 1, 2 y 27. The points of intersection are given by: y2 y A 1 2 3 y 2 y2 3 3 y 0 when y fy g y dy y2 y 2 dy (−3, 3) 3 2y 2 1 gy y 2 y2 2 0 when y f y dy y2 y3 3 2 1 yy A 0, 3 0 dy 9 2 2 1 (4, 2) 3 2y 0 x 1 2 3 4 5 y dy y 1 3 2y 1 3y 0 (1, 3 1) 32 y 2 13 y 3 3 0 9 2 1 (0, 0) −3 −2 −1 x 1 6 Chapter 7 2 Applications of Integration 29. A 1 2 fy y2 1 g y dy 3 y (0, 2) (5, 2) 1 2 0 dy 6 1 x y3 3 2 3 4 5 6 y 1 (0, −1) 2 (2, −1) 3 30. A 0 3 0 fy y 16 3 g y dy 4 y y2 y2 3 0 dy 12 3 ( 3 ,3 7 ) 1 2 2 16 0 2y d y 1 16 y2 0 4 7 1.354 −1 x 1 2 3 31. y A 10 ⇒x x 10 2 10 y 12 1 y 32. A 0 4 4x 4 4 2 4 ln 2 x y dx (1, 4) 1 10 dy y 10 2 (0, 10) 8 6 4 (1, 10) x 0 3 10 ln y 10 ln 10 10 ln 5 33. (a) ln 2 16.0944 −4 4 ln 2 1 (0, 2) (0, 2) −2 (5, 2) x 2 4 6 8 1.227 −1 x 1 3 11 (b) The points of intersection are given by: (3, 9) x3 xx 12 3x 2 1x fx 3x 3 x2 0 when x 3 −6 (0, 0) −1 (1, 1) 0, 1, 3 f x dx 3 1 A 0 1 g x dx 1 gx x2 dx (c) Numerical approximation: 0.417 2.667 3.083 x3 0 1 3x 2 4x 2 43 x 3 3x 3x d x x2 1 x3 3x 2 3x d x 3 x3 0 x3 1 4x 2 32 x 2 3x d x 3 1 x4 4 34. (a) (−1, 2) y 32 x 2 1 0 x4 4 43 x 3 5 12 8 3 37 12 (b) The point of intersection is given by: x3 (1, 0) 2x x3 1 1 2x 0 when x g x dx 1 −2 −1 −1 −2 x 2 1 A 1 1 fx x3 1 1 (1, −2) 2x 1 dx 1 x4 4 2x d x 1 (c) Numerical approximation: 2.0 x3 1 x 1 2 S ection 7.1 35. (a) (0, 3) −6 −3 Area of a Region Between Two Curves 7 9 36. (a) (−2, 8) (4, 3) 12 −4 10 (2, 8) 4 −2 (0, 0) (b) The points of intersection are given by: x2 4x 2x x 4 (b) The points of intersection are given by: x4 2x 2 4 2 3 4 3 3 4x x2 0, 4 x2 4x 3 dx 2x 2 0 when x 2x 2 x4 x4 dx x5 5 2 0 0 when x 4x x2 8x d x 4 x2 x2 A 2 0, ± 2 A 0 4 2x 2 d x 0 2 2x 2 0 2 0 4x 2 4x 3 3 2x 3 3 4x 2 0 64 3 2 128 15 (c) Numerical approximation: 21.333 (c) Numerical approximation: 8.533 (b) The points of intersection are given by: x4 4x 2 4 1 x2 0 0 when x ± 2, ± 1 37. (a) f x x4 2 4x 2, g x x2 4 4 −4 (− 2, 0) (2, 0) 4 x4 x2 4 5x 2 x2 (−1, − 3) −5 (1, − 3) By symmetry: 1 2 (c) Numerical approximation: 5.067 2.933 8.0 A 2 0 1 x4 x4 0 4x 2 5x 2 x2 4 dx 1 4 dx 2 2 1 x2 5x 2 2 4 4 dx x4 4x 2 d x 2 2 2 x5 5 1 5 2 1 x4 x5 5 40 3 5x 3 3 8 5x 3 3 5 3 4 4x 0 2 2 32 5 4x 1 1 5 5 3 4 8 38. (a) f x x4 y 4x 2, g x x3 4x (b) The points of intersection are given by: x4 x4 xx x3 1x 0 4x 2 4x 2 x4 x3 0 4x g4 3 4x 2 2x 0 when x 1 2, 0, 1, 2 x4 0 2 −4 −3 −1 x 1 3 4 A −3 −4 x3 2 4x 4x 2 d x 4x 2 4x x3 x4 4x 4x2 dx dx f x3 (c) Numerical approximation: 8.267 0.617 0.883 9.767 1 248 30 37 60 53 60 293 30 8 Chapter 7 Applications of Integration 3 39. (a) 40. (a) 3 (−1, 1 ( ( 1, 1 ( 2 2 −3 −1 3 −1 −1 ( 3, 9 ) 5 (0, 0) 5 (b) The points of intersection are given by: 1 1 x4 x2 x2 x2 2 1 x 1 3 (b) A 0 6x x2 1 1 0 dx 3 x2 2 0 3 ln x 2 3 ln 10 0 2 x2 0 ±1 6.908 (c) Numerical approximation: 6.908 A 2 0 1 fx 1 1 g x dx x2 dx 2 x3 6 2 1 0 2 0 x2 2 arctan x 2 1 6 4 1 3 1.237 (c) Numerical approximation: 1.237 41. (a) (0, 2) −4 5 42. (a) (2, 3) 2 (0, 1) −1 −1 5 −1 5 (b) and (c) 1 x3 ≤ 1 x 2 2 on 0, 2 (b) and (c) You must use numerical integration: 4 You must use numerical integration because y 1 x3 does not have an elementary antiderivative. 2 A 0 x 4 4 x dx x 3.434 A 0 1 x 2 2 1 x3 dx 1.759 3 6 43. A 2 0 3 fx g x dx 4 3 y 44. A g 2 cos 2x 1 sin 2x 2 sin x dx 6 y 2 0 2 sin x 2 cos x ln 2 tan x d x 3 2 1 π , 3 (π , 1 ) 62 2 −π 2 3 f cos x 3 2 0 π 6 −1 x 2 21 ln cos x 0 π 2 (0, 0) π 2 x 3 4 33 4 0.614 3 4 (− π , −1) 2 π , 3 3 1.299 S ection 7.1 2 1 Area of a Region Between Two Curves x x tan dx 4 4 x 4 1 9 45. A 0 2 2 2 0 cos x cos x dx 2 cos x dx 46. A 0 2 2 42 x 2 4x 4 4 sec 1 4x 4 sec 0 2x y 3 sin x 0 4 12.566 2 2 24 2 2 4 4 1 2 2 4 2.1797 (0, 1) 2 g (2π, 1) 4 y f −1 π 2 π 2π x 3 2 1 x 1 (1, 2) 1 47. A 0 xe 1 e 2 y x2 0 dx 1 0 48. From the graph we see that f and g intersect twice at x 0 and x 1. 1 e 0.316 1 x2 1 1 2 A 0 1 gx 2x 0 f x dx 3 y (1, 3) 1 3x dx 2 1 x2 x 1 ln 3 () (0, 0) 1 1 1, e 1x 3 ln 3 1 (0, 1) 0 −1 x 1 2 x 21 0.180 49. (a) 3 (b) A 0 2 sin x 2 cos x sin 2x dx 1 cos 2x 2 2 1 2 (c) Numerical approximation: 4.0 0 0 0 2 1 2 4 50. (a) − 2 (0, 1) (π, 1) 5 4 (b) A 0 2 sin x 2 cos x cos 2x dx 1 sin 2x 2 4 0 (c) Numerical approximation: 4 4 −2 3 51. (a) 4 (b) A 1 1 1x e dx x2 3 (c) Numerical approximation: 1.323 (1, e) e (3, 0.155) 0 0 6 1x 1 e e1 3 10 Chapter 7 Applications of Integration 5 52. (a) 0 2 (b) A (5, 1.29) 6 1 4 ln x dx x 5 2 1 (c) Numerical approximation: 5.181 (1, 0) 2 ln x 2 ln 5 2 −2 53. (a) 6 (b) The integral 3 (c) A x 4 3 4.7721 A 0 −1 −1 4 x dx does not have an elementary antiderivative. 54. (a) 4 (b) The integral (1, e) 1 (c) 1.2556 A 0 −1 −1 2 x e x dx does not have an elementary antiderivative. 55. (a) 5 56. (a) 4 −3 −1 3 −4 −1 3 (b) The intersection points are difficult to determine by hand. c (b) The intersection points are difficult to determine. (c) Intersection points: 1.164035, 1.3549778 and 1.4526269, 2.1101248 1.4526269 (c) Area c 4 cos x x2 dx 6.3043 where c 1.201538. A 1.164035 3 x x 2 dx 3.0578 x 57. F x 0 1 t 2 0 1 dt t2 4 x t 0 x2 4 x 22 4 y (a) F 0 y (b) F 2 2 3 (c) F 6 y 6 5 4 3 2 62 4 6 15 6 5 4 3 2 6 5 4 3 2 −1 −1 t 1 2 3 4 5 6 −1 −1 t 1 2 3 4 5 6 −1 −1 t 1 2 3 4 5 6 S ection 7.1 x Area of a Region Between Two Curves 11 58. F x 0 12 t 2 0 2 dt 13 t 6 x 2t 0 x3 6 2x 43 6 24 56 3 (c) F 6 y y (a) F 0 y (b) F 4 36 12 48 20 20 16 16 12 12 8 8 4 4 x 1 2 3 4 5 6 1 2 3 4 5 6 x 20 16 12 8 4 x 1 2 3 4 5 6 59. F 1 cos 1 0 y 2 d 2 sin 2 2 1 sin 2 2 2 y (a) F (b) F 0 0.6366 (c) F 1 2 2 y 3 2 2 1.0868 3 2 3 2 1 2 1 2 1 2 1 2 1 −2 1 θ 1 −2 1 2 1 −2 1 θ 1 −2 1 −2 1 −2 1 2 1 θ y y 60. F y 1 4ex 2 dx 1 y 30 25 20 15 10 5 −1 8ex 2 1 8e y 2 8e 12 (a) F 0 (b) F 0 y 30 25 20 15 10 5 x 8 8e 12 3.1478 (c) F 4 y 30 25 20 15 10 5 8e2 8e 12 54.2602 1 2 3 4 −1 x 1 2 3 4 −1 1 2 3 4 x 4 61. A 2 4 2 9 x 2 7 x 2 6 12 7 dx x 6 4 5 dx 4 5 x 2 16 x 5 dx y 6 9 y = 2 x − 12 7 x 2 21x (4, 6) 5 y = − 2 x + 16 21 d x 6 4 2 72 x 4 4 7x 2 72 x 4 7 4 7 14 2 −2 −4 6 (6, 1) x 8 10 (2, −3) y=x−5 12 Chapter 7 c Applications of Integration a c b y dy c 62. A 0 c 0 b a y c a2 y 2c y a 63. 4 3 2 1 y a dy c −4 −2 −1 (0, 2) (4, 2) x 2 3 4 ay 0 (− 4, − 2) −3 −4 (0, − 2) ac 2 y ac ac 2 1 base height 2 Left boundary line: y Right boundary line: y 2 x x 2⇔x 2⇔x y y 2 2 y= cx b (b, c) A y= c b − a (x − a ) 2 2 y 4 dy 2 2 2 y 8 2 2 dy 8 16 4y x (0, 0) (a, 0) 1 3 64. A 0 1 2x 5x dx 0 3x dx 1 3 1 2x 15 dx 2 3 1 4 1 x 2 7 2 dx y 2 1 5 x 2 15 x 2 5 4 (1, 2) (0, 0) −2 −1 −1 −2 −3 2 3 4 x 5x2 2 5 2 15 2 1 0 5x2 4 45 4 45 2 (3, −2) (1, −3) 15 2 −4 65. Answers will vary. If you let x (a) Area 60 0 2 10 3 322 (b) Area 60 0 3 10 2 502 66. x 4, n 8, b 32 0 28 2 190.8 (b) Area 32 0 38 4 296.6 3 a 2 14 966 sq ft 4 14 1004 sq ft 84 2 11 6 and n 2 14 2 12 10, b 2 12 a 10 6 2 15 60. 2 20 2 23 2 25 2 26 0 2 14 4 12 2 12 4 15 2 20 4 23 2 25 4 26 0 32 2 13.5 2 14.2 2 14 2 14.2 2 15 2 13.5 0 (a) Area 381.6 sq mi 4 11 2 13.5 4 14.2 2 14 4 14.2 2 15 4 13.5 0 395.5 sq mi S ection 7.1 67. f x fx x3 3x 2 3. 1 3x 1 or y x 3 at x x4 4 3x 2 2 3x 2 2. 1 −4 −3 −2 8 6 4 2 y Area of a Region Between Two Curves 13 At 1, 1 , f 1 Tangent line: y y = 3x − 2 (1, 1) x 1 2 3 4 The tangent line intersects f x 1 f (x) = x3 −6 A 2 x3 3x 2 dx 2x 2 27 4 (− 2, − 8) −8 68. y y y 1 x3 3x 2 3 2x, 2 2 1 1 1, 1 5 4 y (2, 4) y=x+2 3 Tangent line: y 1x 1 ⇒y (− 1, 1) 2 x −1 −2 −3 2 3 4 x 2 −4 Intersection points: 2 1, 1 and 2, 4 2 y = x 3 − 2x A 1 x x4 4 1 x2 x2 2 3x2 2 x3 2 2x dx 1 x3 4 6 4 3x 1 4 2 dx 3 2 2 27 4 2x 1 69. f x fx y 1 2x 1 2 3 4 1 2 1 4 (0, 1) 1 f ( x) = 2 x +1 1 At 1, , f 1 2 Tangent line: y 1 2 1 . 2 1 x 2 1 or y 1 x2 1 dx 1 at x arctan x 1 x 2 0. x2 4 1 (1, 21 ) y=− 1x+1 2 x 1 2 1 1 3 2 2 The tangent line intersects f x 1 A 0 1 x2 2 1 1 8 22 4x2 1 1 x 2 1 ,1 2 x 0 3 4 0.0354 70. y y 1 y 2 , y 16x 4x2 y = − 2x + 2 2 1 (0, 2) y = 2 1 + 4x 2 ( 1 , 1( 2 x 1 2 1 y 2x 2x 2 1 2 −1 Tangent line: y Intersection points: 12 1 , 1 , 0, 2 2 12 2 A 0 2 1 4x 2x 2 dx arctan 2x x2 2x 0 arctan 1 1 4 1 4 3 4 0.0354 14 71. x 4 A Chapter 7 2x 2 1 Applications of Integration x 2 on x4 1, 1 2x 2 1 dx y 1≤1 1 x2 x 4 dx x5 1 5 1 72. x 3 ≥ x on 1, 0 , x 3 ≤ x on 0, 1 Both functions symmetric to origin. 0 1 1 1 x3 x2 1 1 x dx 0 x3 x dx x3 dx 0. x dx y 1 x 3 3 4 15 Thus, 2 x3 1 1 (0, 1) 1 (1, 1) You can use a single integral because x 4 2x 2 1 ≤ 1 x 2 on 1, 1 . A x 2 0 x (0, 0) ( 1, 0) (1, 0) x2 2 2 x4 4 1 0 1 2 −1 x 1 −1 (− 1, − 1) 73. Offer 2 is better because the accumulated salary (area under the curve) is larger. 3 74. Proposal 2 is better since the cumulative deficit (the area under the curve) is less. 75. 9 9 9 0 A 3 b 9 9 x2 b x3 3 2 9 3 9 x2 dx b dx x2 dx 9 0 32 b 36 10 y 18 9 6 2 b b 6 4 9 9 bx 2 x 2 6 ( 9 b, b) ( 9 b, b) 9 b b 9 9 27 2 3 32 b 9 4 9 4 b 9 3 3.330 9 76. A 2 0 9 2 x dx 9 0 9 b b 2 9x 9 9 x b bx 9 x2 2 b9 9 b 9 x2 2 9 81 0 12 y b dx x dx 9 0 b 81 2 81 2 81 2 81 2 9 2 2.636 9 (− (9 − b), b) 6 (9 −b, b) x 3 6 2 0 −6 −3 −3 −6 29 b b 9 2 S ection 7.1 1 77. Area of triangle OAB is 2 4 4 a Area of a Region Between Two Curves 15 8. 4x x2 2 a y 4 0 4 0 x dx 4a 0 a2 2 B 3 2 1 A a 2 8a 8 a 4±2 2 4 22 2 a x 1 2 3 4 O Since 0 < a < 4, select a 2 1.172. 78. Total area 2 4 2 4y y 2 dy y3 3 2 2 0 4 8 3 y 2 dy 3 y 28 0 32 3 4 1 16 3 4 42 3 4 2 a 4 a a 42 3 32 x dx 4 4 3 x 32 a 4 4 3 −1 x −1 1 2 3 4 5 a 32 a −3 4 4 4 n a 1.48 n 79. lim →0 i xi 1 x i2 x x 1 is the same as n x3 3 1 0 80. lim →0 i 4 1 x i2 2 x 4i and n 4x x x3 3 2 2 where xi 1 i and n x2 dx where xi 2 4 is the same as n 32 . 3 x 0 y x2 2 1 . 6 4 2 x2 dx y 5 0.6 0.4 0.2 0.2 0.4 f (x) x x2 3 f ( x) = 4 − x 2 (1, 0) x 2 0.6 0.8 1.0 (0, 0) (− 2, 0) −3 −1 1 1 −1 (2, 0) x 3 5 5 81. 0 7.21 0.58t 7.21 0.45t d t 0 0.13t d t 0.13 t 2 2 5 5 $1.625 billion 0 5 82. 0 7.21 0.26t 0.02t 2 7.21 0.1t 0.01t 2 d t 0 0.01t 2 0.01t 3 3 29 billion 12 0.16t d t 0.16t 2 2 5 0 $ 2.417 billion 16 Chapter 7 Applications of Integration t 83. (a) y1 Receipts (in billions) 600 500 400 300 200 100 270.3151 1.0586 R 270.3151e0.05695t (b) y2 Expenditures (in billions) 600 500 400 300 200 100 239.9704 1.0416 E t 239.9704e0.04074t t 2 4 6 8 10 12 t 2 4 6 8 10 12 Time (in years) 17 Time (in years) (c) Surplus 12 y1 y2 dt 926.4 billion dollars (Answers will vary.) 84. (a) y1 (b) Percents of total income 100 80 60 40 20 x 20 40 60 80 100 (d) No, y1 > y2 forever because 1.0586 > 1.0416. No, these models are not accurate for the future. According to news, E > R eventually. 0.0124x 2 y 0.385x 7.85 (c) Percents of total income 100 80 60 40 20 y x 20 40 60 80 100 Percents of families 100 Percents of families (d) Income inequality 0 x y1 dx 2006.7 85. 5%: P1 31%: P2 2 893,000e 0.05 t 893,000e 0.035 t 5 Difference in profits over 5 years: 0 893,000e 0.05t 893,000e 0.035t d t 893,000 e 0.05t e 0.035t 5 0.05 0.035 0 893,000 25.6805 34.0356 $ 193,156 20 28.5714 893,000 0.2163 Note: Using a graphing utility, you obtain $193,183. 86. The total area is 8 times the area of the shaded region to the right. A point x, y is on the upper boundary of the region if 2 y x2 x2 y2 y2 x2 4y y 2 4 4 4 1 y 4y 4y x2 x2 . 4 x. y2 1 y=x ( x, y ) x 1 2 We now determine where this curve intersects the line y x x2 4x 4 x Total area 8 0 1 0 x2 4 4± 2 2 16 2 2 16 x2 4 2±2 2 ⇒ x x dx 8x x3 12 x2 2 2 2 0 2 22 2 1 16 42 3 5 8 0.4379 3.503 S ection 7.1 87. The curves intersect at the point where the slope of y2 equals that of y1, 1. y2 0.08x2 k ⇒ y2 0.16x 1⇒x 1 0.16 6.25 Area of a Region Between Two Curves 17 6.25 (a) The value of k is given by y1 6.25 k y2 0.08 6.25 3.125. 2 (b) Area 2 0 6.25 y2 y1 d x 3.125 x2 2 x dx 6.25 0 k 2 0 0.08x 2 0.08x 3 3 2 3.125x 2 6.510417 5 13.02083 1 16 2 88. (a) A 2 0 1 x 2 5 9 1 3 5.5 5 x 32 x dx 5 5 5.5 1 x 0 dx 89. (a) A (b) V 6.031 2A 2 2 5.908 2 11.816 m 3 1 8 2 5.908 2 0 5 (c) 5000V 5000 11.816 59,082 pounds 25 (b) V 2A 10 5 9 2 6.031 5.5 5 6.031 m 2 12.062 m 3 60,310 pounds 91. True 3 7 6 (c) 5000 V 90. True 5000 12.062 92. False. Let f x x and g x 2x x2. f and g intersect at 1, 1 , the midpoint of 0, 2 . But b 2 93. Line: y 7 x 1 y fx a g x dx 0 x 2x x2 dx 2 3 0. A 0 sin x cos x 3 2 2.7823 7 24 3x2 14 1 3x dx 7 7 0 6 1 2 (0, 0) π 6 −1 4π 3 x ( 76π , − 1 ( 2 a 94. A a 4 0 b 1 x2 dx a2 4b a a a2 0 x2 dx a2 . 4 y y=b a2 0 x2 dx is the area of 4b a a2 4 1 of a circle 4 x 1− 2 a 2 b x Hence, A ab. a 18 Chapter 7 Applications of Integration y 95. We want to find c such that: b y = 2x − 3x 3 2x 0 3x 3 c dx b 0 c (b, c) x2 b2 But, c b2 4 34 4b 34 4x 34 4b cx 0 0 x cb 0 2b 3b3 because b, c is on the graph. 2b 8 3b3 b 12b2 9b 2 0 0 4 2 3 4 9 3b2 b c Section 7.2 1 Volume: The Disk Method 1 1. V 0 x 1 2 dx 0 x2 2x 1 dx x3 3 1 x2 x 0 3 2 2 2. V 0 4 x2 2 dx 0 x4 8x 2 16 dx x5 5 8x 3 3 2 16x 0 256 15 3 3 4 4 3. V 1 x 2 dx 1 x dx x2 2 4 1 15 2 4. V 0 9 x2 2 dx 0 9 9x x2 dx x3 3 3 18 0 1 1 5. V 0 x2 2 x3 2 dx 0 x4 x6 dx x5 5 x7 7 1 0 2 35 6. 2 8 x2 x 4 16 8 ±2 x2 4 x2 2 2 V 2 2 2 2 4 2 0 x2 4 2 2 2 dx 7. y V x2 ⇒ x 4 y 4 x4 16 2x3 3 2x2 2 12 dx 2 y 0 2 dy 0 y dy 2 2 2 x5 80 12x 0 y2 2 4 8 0 128 2 80 32 2 3 132.69 24 2 448 2 15 8. y V 0 16 4 x2 ⇒ x 16 y2 4 0 2 16 dy y2 4 9. y 16 y2 dy V x2 3 ⇒x 1 y3 2 1 y3 2 2 dy 0 0 y3 dy 0 y4 4 1 0 4 16y y3 3 128 3 S ection 7.2 4 4 Volume: The Disk Method 19 10. V 1 y2 y5 5 2y 4 4y 2 d y 1 y4 4 1 8y 3 153 5 16y 2 d y 16y 3 3 4 0 2 459 15 11. y x, y 0, x x, r x 4 (a) R x V 0 4 (b) R y V 4, r y 2 y2 y4 dy 15 y 5 2 0 x x dx 0 y dx 4 16 0 2 x2 0 8 y 3 2 1 x 16y 128 5 3 2 1 x 1 −1 2 3 4 1 −1 2 3 4 (c) R y V 4 2 y 2, r y 4 y2 2 dy 8y 2 83 y 3 0 (d) R y V 6 2 y 2, r y 6 y2 2 2 4 dy y4 dy 15 y 5 2 0 0 2 0 2 16 0 y4 dy 0 32 256 15 y 4 3 12y 2 4y 3 16y y 3 2 15 y 5 2 0 32y 192 5 2 1 x 1 −1 2 3 −1 −2 1 x 1 2 3 4 5 12. y 2x 2, y 0, x 2, r y 8 2 y2 y dy 2 4y y 4 28 (a) R y V (b) R x 16 0 2x 2, r x 2 0 4x 5 5 2 0 4 0 y 8 6 V 0 4x 4 d x y 8 6 128 5 4 2 x 2 4 −4 −2 4 2 x 2 4 −4 −2 —CONTINUED— 20 Chapter 7 Applications of Integration 12. —CONTINUED— (c) R x V 0 2 2 8, r x 2 8 64 2x 2 32 x 2 4 0 (d) R y 4 x4 dx 8x 2 y 2 8 y 2, r y 2 y 2 4 y 2 2 0 dy y dy 2 y2 4 8 y 64 32x 2 0 V 0 8 4x 4 d x 15 x 5 2 0 x4 dx 0 4 4y 16 3 4 896 15 83 x 3 4 2 32 y 3 0 8 6 6 4 2 x 2 4 −4 −2 4 2 x 4 −4 −2 13. y x2, y 4x 4x 2 x 2 intersect at 0, 0 and 2, 4 . x 2, r x 4x x2 2 (a) R x V x2 x 4 dx (b) R x V 6 2 x2, r x 6 x2 2 6 6 4x 4x x2 x2 2 dx 0 2 0 2 16x 2 0 8x 3 d x 2 8 0 x3 x4 4 y 5x 2 53 x 3 6x d x 2 16 3 x 3 y 4 2x 4 0 32 3 8 3x 2 0 64 3 5 3 4 2 1 1 −1 x 1 2 3 −2 −1 1 2 3 4 x 3 2 14. y 6 2x 6 0 x 2, y 2x 6 3 0 x 6 intersect at x 2 3, 3 and 0, 6 . (b) R x 2 (a) R x V x 2, r x 2x 4x 3 x4 y 8 6 x 6 dx 6 0 2x 3 2x x2 x2 3, r x 2 x 3 2 6 dx 3 x2 V 3 0 x x4 3 9x 2 3x 3 36x d x 3 0 x4 243 5 15 x 5 4x 3 x4 y 8 3x 2 x3 9x 2 18x d x 0 3 15 x 5 18x 2 3 108 5 4 2 x 2 4 2 x 2 −6 −4 −2 −6 −4 −2 S ection 7.2 x3 ,r x 2 4 0 2 Volume: The Disk Method 21 15. R x V 4 3 x, r x 4 x 2 1 1 2 16. R x dx y 5 4 1 0 2 0 3 V x3 2 dx x6 dx 4 x7 28 128 28 2 y x2 0 8x 4x2 15 d x 3 16 0 4x3 x4 16 x3 3 18 15x 0 3 2 1 x 1 2 3 4 16x 32 144 7 0 3 2 1 −1 −1 (2, 4) −1 x 1 2 3 4 17. R x V 4, r x 3 4 4 8 1 1 1 1 1 1 x 1 1 1 3 x 2 18. R x dx dx 3 4, r x 3 4 2 sec x 4 sec x 2 42 0 3 0 V 0 3 4 dx x 2 8 sec x 0 sec 2 x d x 3 1 x x 1 4 3 4 8 ln 1 8 ln 4 8 ln 4 32.485 8 ln sec x 8 ln 2 y tan x 3 3 3 3 tan x 0 x 0 8 ln 1 27.66 0 0 8 ln 2 y 5 2 1 −1 −1 x 1 2 3 4 2 1 π 9 π 3 x 3 2π 9 4π 9 5π 9 19. R y V 6 4 y, r y 6 y 2 dy 12y 6y 2 0 20. R y V 6, r y 4 6 y y3 3 4 0 2 6 dy 368 3 y y 6 0 2 0 4 y2 0 36 d y 4 y y 5 5 4 4 3 3 2 2 1 1 36y y3 3 208 3 36y 0 x x 1 −1 2 3 4 5 −1 1 2 3 4 5 22 Chapter 7 Applications of Integration 6 ,ry y 6 2 6 21. R y V 6 2 y 2, r y 6 y2 2 2 2 2 22. R y dy V 36 6 6 0 dy 2 2 6 y 1 2 2 0 y4 y5 5 12y 2 4y 3 32y 32 dy 2 y 2 2 y 2 ln y 1 dy y2 1 y 6 2 2 384 5 0 4 3 2 1 −1 −2 −3 x 1 2 3 5 36 36 36 y 35 6 13 3 2 ln 6 2 ln 1 3 3 2 y 6 5 4 3 2 1 2 ln 2 12 13 241.59 6 ln 3 x 1 2 3 4 5 23. R x 3 1 x x 1 x 1 1 1 , rx 2 0 dx 2 y 24. R x V 2 x4 2 x 2, r x x2 2 0 dx 3 2 1 −3 −1 x 1 2 3 y V 0 3 0 x4 0 2 1 dx 2 0 1 4x 2 4x 3 3 x4 dx x5 2 5 0 3 ln x ln 4 1 0 1 −1 2 3 x 2 128 15 −2 −3 4.355 25. R x V 1 , rx x 4 1 0 2 y 26. R x V 3 x 8 0 1 x 8 , rx 3 1 1 2 0 4 y 1 x 1 x 4 1 2 dx 1 x 1 −1 −2 2 3 4 dx 2 3 2 9 0 x 1 x dx 1 3 4 e x, r x 1 8 9 1 8 0 x 2 4 6 8 27. R x V 0 2 y 28. R x V e x 2, r x 4 0 8 y e 0 1 x2 dx 1 ex 0 4 22 dx 6 4 2 e 0 2x dx 1 ex dx 0 4 x 2 2 1 ...
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