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Unformatted text preview: 4 37 8 21 4 47 8 26 4 57 8 31 4 67 8 3 12.6667 2 9. Exact: 1 2 1 x 1 x 1 2 1 2 dx dx 1 x 11 84 11 84 1 2 32 81 4 2 1 1 3 1 54 8 25 1 54 8 25 1 1 2 2 1 6 2 1 9 2 1 9 0.1667 1 32 1 2 Trapezoidal: 1 2 1 74 1 2 1 9 32 121 1 2 0.1676 1 32 1 2 2 Simpson’s: 1 1 x 1 2 dx 11 12 4 11 12 4 4 1 74 1 2 1 9 64 81 64 121 0.1667 2 11. Trapezoidal: 0 2 1 1 0 x3 dx x3 dx 1 1 4 1 1 6 21 41 18 18 22 22 21 41 27 8 27 8 3 3 3.283 3.240 Simpson’s: Graphing utility: 3.241 1 1 13. 0 x1 x dx 0 1 x1 x dx 1 0 8 1 0 12 1 1 4 1 1 4 1 4 1 4 1 1 2 1 1 2 1 2 1 2 3 1 4 3 1 4 3 4 3 4 Trapezoidal: 0 x1 x dx 2 2 2 0.342 1 Simpson’s: 0 x1 x dx 4 2 4 0.372 Graphing utility: 0.393 206 Chapter 4 Integration 2 15. Trapezoidal: 0 cos x2 dx 2 8 0.957 cos 0 2 cos 2 4 2 2 cos 2 2 2 2 cos 2 4 2 2 cos 2 2 Simpson’s: 0 cos x2 dx 2 12 0.978 cos 0 4 cos 2 4 2 2 cos 2 2 2 4 cos 2 4 2 2 cos 2 Graphing utility: 0.977 1.1 17. Trapezoidal: 1 1.1 sin x2 dx sin x2 dx 1 1 sin 1 80 1 sin 1 120 2 sin 1.025 2 2 sin 1.05 2 2 2 sin 1.075 2 2 sin 1.1 2 2 0.089 2 Simpson’s: 4 sin 1.025 2 sin 1.05 4 sin 1.075 sin 1.1 0.089 Graphing utility: 0.089 4 19. Trapezoidal: 0 4 x tan x dx x tan x dx 0 32 48 0 0 2 4 16 16 tan tan 16 16 2 2 2 2 tan 16 16 2 2 tan 16 16 2 4 3 3 tan 16 16 3 3 tan 16 16 4 4 0.194 0.186 Simpson’s: Graphing utility: 0.186 x3 3x2 6x 6 0 20 12 42 3 21. (a) y 23. y = f ( x) fx fx fx fx f 4 x x a b (a) Trapezoidal: Error ≤ 12 0.5 since 2. The Trapezoidal Rule overestimates the area if the graph of the integrand is concave up. f x is maximum in 0, 2 when x (b) Simpson’s: Error ≤ f 4 20 0 180 44 5 0 since x 0. 25. f x 2 in 1, 3 . x3 1 and f 1 2. 366. (a) f x is maximum when x Trapezoidal: Error ≤ f (b) f 4 4 23 12n2 2 < 0.00001, n2 > 133,333.33, n > 365.15; let n x 24 in 1, 3 x5 1 and when f 4 x is maximum when x 25 180n4 1 24. 26. Simpson’s: Error ≤ 24 < 0.00001, n4 > 426,666.67, n > 25.56; let n S ection 4.6 Numerical Integration 207 27. f x (a) f x 1 x 1 41 x 32 in 0, 2 . 0 and f 0 1 . 4 130. f x is maximum when x Trapezoidal: Error ≤ (b) f 4 81 12n2 4 in 0, 2 < 0.00001, n2 > 16,666.67, n > 129.10; let n x 4 16 1 15 x 72 f x is maximum when x 32 15 180n4 16 0 and f 4 0 15 . 16 12. Simpson’s: Error ≤ < 0.00001, n4 > 16,666.67, n > 11.36; let n 29. f x (a) f x tan x2 2 sec2 x2 1 4x2 tan x2 in 0, 1 . 1 and f 1 3 f x is maximum when x Trapezoidal: Error ≤ (b) f 4 49.5305. 643. 10 12n2 3 49.5305 < 0.00001, n2 > 412,754.17, n > 642.46; let n 32x4 tan x2 1 and f 4 x 4 8 sec2 x2 12x2 36x2 tan2 x2 9184.4734. 48x4 tan3 x2 in 0, 1 f x is maximum when x 1 Simpson’s: Error ≤ 1 05 9184.4734 < 0.00001, n4 > 5,102,485.22, n > 47.53; let n 180n4 Cx 5 48. 31. Let f x Ax3 Bx2 D. Then f 0 4 x 0. Simpson’s: Error ≤ ba 0 180n4 Therefore, Simpson’s Rule is exact when approximating the integral of a cubic polynomial. 1 Example: 0 x3 dx 1 0 6 4 1 2 3 1 1 4 This is the exact value of the integral. 33. f x n 4 8 10 12 16 20 2 Ln 12.7771 14.0868 14.3569 14.5386 14.7674 14.9056 3x2 on 0, 4 . Mn 15.3965 15.4480 15.4544 15.4578 15.4613 15.4628 Rn 18.4340 16.9152 16.6197 16.4242 16.1816 16.0370 Tn 15.6055 15.5010 15.4883 15.4814 15.4745 15.4713 Sn 15.4845 15.4662 15.4658 15.4657 15.4657 15.4657 208 Chapter 4 Integration 35. f x n 4 8 10 12 16 20 sin x on 0, 4 . Ln 2.8163 3.1809 3.2478 3.2909 3.3431 3.3734 2 Mn 3.5456 3.5053 3.4990 3.4952 3.4910 3.4888 Rn 3.7256 3.6356 3.6115 3.5940 3.5704 3.5552 Tn 3.2709 3.4083 3.4296 3.4425 3.4568 3.4643 Sn 3.3996 3.4541 3.4624 3.4674 3.4730 3.4759 37. A 0 x cos x dx 14 84 0.701 y Simpson’s Rule: n 2 x cos x dx 0 0 cos 0 4 28 cos 28 2 14 cos 14 4 3 3 cos 28 28 ... 2 cos 2 1 1 2 π 4 π 2 x 5 39. W 0 100x 125 x 3 dx 12 x3 dx 5 0 3 12 400 400 5 12 15 12 125 125 5 12 15 12 3 Simpson’s Rule: n 5 100x 125 0 200 3 10 12 0 125 10 12 3 ... 10,233.58 ft lb 12 41. 0 6 1 x2 dx Simpson’s Rule, n 6 1 0 2 36 6 4 6.0209 3.1416 2 6.0851 4 6.1968 2 6.3640 4 6.6002 6.9282 1 113.098 36 43. Area 1000 125 2 10 2 125 2 120 2 112 2 90 2 90 2 95 2 88 2 75 2 35 89,250 sq m t 45. 0 sin x dx 2, n 10 2.477. By trial and error, we obtain t...
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