Explanation Calculate the exact value of the integral and obtain the end points. Compute T N for various values of N = 4 , 8 , 16 , 32 , 64 and compare with the previous exercise to verify the given statement Given: Observe that the error bound for T N (which has 12 in the denominator) is twice as large as the error bound for M N (which has 24 in the denominator). Calculation: The exact value of the integral is ∫ 0 π sin x d x = [ − cos x ] 0 π = 2 Δ x = π 4 and the endpoints are 0 , π 4 , 2 π 4 , 3 π 4 , π Hence, T 4 = 1 2 π 4 [ sin 0 + 2 sin ( π 4 ) + 2 sin ( 2 π 4 ) + 2 sin ( 3 π 4 ) + sin π ] ≈ 1.896119 Similarly, T 8 = 1 2 π 8 [ sin 0 + 2 sin ( π 8 ) + 2 sin ( 2 π 8 ) + ... + sin π ] ≈ 1.974232 T 16 = 1 2 π 16 [ sin 0 + 2 sin ( π 16 ) + 2 sin ( 2 π 16 ) + ... + 2 sin ( 15 π 16 ) + sin π ] ≈ 1.993570 T 32 = 1 2 π 32 [ sin 0 + 2 sin ( π 32 ) + 2 sin ( 2 π 32 ) + ...

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

3rd Edition

ISBN: 9781319048518

Author: Colin Adams Jon Rogawski

Publisher: Macmillan Higher Education

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Question

Chapter 7.9, Problem 55E

To determine

The actual error in $T_{N}$ for $\int_{0}^{π} \sin x d x$ for $N = 4, 8, 16, 32, 64$ and compare it with the calculations of the previous exercise. Determine if the actual error in $T_{N}$ seem to be roughly twice as large as the error in $M_{N}$ in this case

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Chapter 7.9, Problem 54E

Chapter 7.9, Problem 56E

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Chapter 7 Solutions

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

Ch. 7.1 - Prob. 1PQ Ch. 7.1 - Prob. 2PQ Ch. 7.1 - Prob. 3PQ Ch. 7.1 - Prob. 1E Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E

Knowledge Booster

The actual error in T N for ∫ 0 π sin x d x for N = 4 , 8 , 16 , 32 , 64 and compare it with the calculations of the previous exercise. Determine if the actual error in T N seem to be roughly twice as large as the error in M N in this case

The actual error in TNfor ∫0πsinxdx for N=4,8,16,32,64 and compare it with the calculations of the previous exercise. Determine if the actual error in TNseem to be roughly twice as large as the error in MN in this case

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Chapter 7 Solutions

The actual error in $T_{N}$ for $\int_{0}^{π} \sin x d x$ for $N = 4, 8, 16, 32, 64$ and compare it with the calculations of the previous exercise. Determine if the actual error in $T_{N}$ seem to be roughly twice as large as the error in $M_{N}$ in this case