# 464 s ection 43 riemann sums and definite integrals

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Unformatted text preview: n 1 6n2 2 10 3 1 3 n x2 1 1 dx n→ lim n n 3 9. lim →0 i 3ci 1 10 xi 1 3x 10 dx 11. lim →0 i ci2 1 4 xi 0 x2 4 dx on the interval 5 1, 5 . 4 on the interval 0, 3 . 2 13. 0 3 dx 15. 4 4 x dx 17. 2 4 x2 dx 2 19. 0 sin x dx 21. 0 y 3 dy 23. Rectangle A A 0 y bh 3 34 4 dx 12 5 3 2 1 Rectangle x 1 2 3 4 5 190 Chapter 4 Integration 25. Triangle A A 0 y 1 bh 2 4 1 44 2 8 4 27. Trapezoid b1 b2 h A 2 Triangle 2 y 5 2 5 dx 9 2 9 6 x dx 2 A 0 x 2 4 2x 14 3 Trapezoid x 1 2 3 29. Triangle A A 1 y 31. Semicircle Triangle y 1 bh 2 1 1 21 2 1 x dx 1 1 1 A x 12 r 2 3 1 2 9 4 Semicircle 3 2 1 A 3 x2 dx 9 2 x 4 2 2 4 4 4 4 In Exercises 33–39, 2 x3 dx 60, 2 x dx 6, 2 dx 2 2 4 4 4 33. 4 x dx 2 x dx 6 35. 2 4 4x dx 4 2 x dx 46 4 24 4 4 4 4 4 37. 2 x 8 dx 2 x dx 8 2 dx 6 82 10 39. 2 13 x 2 3x 2 dx 1 2 x3 dx 2 3 2 x dx 22 6 2 2 dx 1 60 2 7 5 7 6 6 36 f x dx 16 41. (a) 0 f x dx 0 0 f x dx 5 5 f x dx 10 10 3 13 43. (a) 2 fx g x dx 2 g x dx 2 (b) 5 5 f x dx 0 f x dx 0 5 10 6 6 2 g x dx 8 6 (b) 2 gx f x dx 2 f x dx 2 (c) 5 f x dx 5 2 3f x dx 3 0 10 2 2 12 4 30 (d) 0 f x dx 3 10 30 6 6 (c) 2 2g x dx 6 2 2 g x dx 6 (d) 2 3f x dx 3 2 f x dx 3 10 45. (a) Quarter circle below x-axis: (b) Triangle: 1 bh 2 (c) Triangle 1 2 1 4 r2 1 4 2 2 42 4 1 2 Semicircle below x-axis: 1 2 21 3 1 20 2 1 2 2 2 1 2 (d) Sum of parts (b) and (c): 4 (e) Sum of absolute values of (b) and (c): 4 (f) Answer to (d) plus 2 10 20: 3 2 2 23 5 2 2 47. The left endpoint approximation will be greater than the actual area: > 49. Because the curve is concave upward, the midpoint approximation will be less than the actual area: < S ection 4.3 Riemann Sums and Definite Integrals 191 51. f x 1 x 4 53. 4 3 2 1 y is not integrable on the interval 3, 5 and f has a discontinuity at x 4. 1 2 3 4 x a. A 5 square units 55. 2 3 2 y 1 1 2 x 1 2 1 3 2 2 1 d. 0 3 2 sin x dx 1 12 2 1 57. 0 x3 n Ln Mn Rn 2 x dx 4 3.6830 4.3082 3.6830 8 3.9956 4.2076 3.9956 12 4.0707 4.1838 4.0707 16 4.1016 4.1740 4.1016 20 4.1177 4.1690 4.1177 59. 0 sin2 x dx n Ln Mn Rn 4 0.5890 0.7854 0.9817 8 0.6872 0.7854 0.8836 12 0.7199 0.7854 0.8508 16 0.7363 0.7854 0.8345 20 0.7461 0.7854 0.8247 61. True 63. True 65. False 2 x dx 0 2 67. f x x0 x1 c1 x2 0, x1 3x, 0, 8 1, x2 3, x3 7, x4 4, x4 8 f2 10 2 x2 f5 40 4 x3 f8 x4 272 1 8 1, x2 1, c2 4 2, x3 2, c3 x 5, c4 f1 41 f ci i 1 x1 88 1 192 Chapter 4 Differentiation 69. f x 1, 0, x is rational x is irrational → 0, f ci 1 or f ci 0 in each subinterval since there is not integrable on the interval 0, 1 . As are an infinite number of both rational and irrational numbers in any interval, no matter how small. 71. Let f x n x2, 0 ≤ x ≤ 1, and xi n 1 n. The appropriate Riemann Sum is f ci i 1 xi i 1 i n 32 2 1 n 1n2 i. n3i 1 ... n2 n→ 1 lim 3 12 n→ n 22 lim lim 1 n3 2n2 n 2n 3n 6n2 1n 6 1 1 1 3 1 2n 1 6n2 1 3 n→ n→ lim Section 4.4 1. f x 4 0 The Fundamental Theorem of Calculus 5 4 x2 x2 1 1 dx is positive. −5 3. f x 2 x x2 x x2 1 0 −5 5 1 dx 5 5 2 −5 −2 1 1 0 5. 0 2x dx x2 0 1 0 1 7. 1 x 2 dx x2 2 0 2x 1 0 1 2 2 5 2 1 9. 1 t2 2 dt t3 3 1 1 2t 1 1 3 2 1 3 43 t 3 2 10 3 1 1 11. 0 2 2t 1 2 dt 0 4t2 4t 1 dt 2t2 t 0 4 3 1 2 2 3 2 1 1 3 13. 1 3 x2 u 1 dx 3 x 4 2 x 1 3 2 2 3 1 4 15. 1 1 2 du u 3 u1 1 2 2u 12 du 23 u 3 3 4 1 x2 32 4 2 4u1 2 1 4 3 44 2 3 4 2 3 17. 1 1 t 2 dt 34 t 4 1 3 1 1 3 2t 1 2 3 4 23 x 3 3 4 1 2 0 2 4 19. 0 0 x 3 x dx x 0 x1 2 dx 11 32 3 5 27 20 2 3 1 18 21. 1 t1 3 t2 3 dt 34 t 4 32 3 35 t 5 0 3 1 0 3 3 23. 0 2x 3 dx 0 3 3x x2 2x dx 32 32 2x 3 3 dx 9 2 split up the integral at the zero x...
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## This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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