Question

54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
59. ∫(from 0 to π) ln(5 + 3cosx) dx = π ln(9/2)

Accepted Answer

Step 1: Understand the problem. You are tasked with comparing the errors in the Midpoint Rule and Trapezoid Rule for approximating the integral ∫(from 0 to π) ln(5 + 3cosx) dx, with the exact value of the integral given as π ln(9/2). You will perform this comparison for n = 4, 8, 16, and 32 subintervals. Step 2: Recall the formulas for the Midpoint Rule and Trapezoid Rule. The Midpoint Rule approximates the integral by summing the function values at the midpoints of subintervals, while the Trapezoid Rule uses the average of the function values at the endpoints of subintervals. Both methods divide the interval [0, π] into n subintervals of equal width Δx = (π - 0)/n. Step 3: For the Midpoint Rule, calculate the midpoints of each subinterval. The midpoints are given by x_i = a + (i - 0.5)Δx, where i ranges from 1 to n. Evaluate the function ln(5 + 3cosx) at each midpoint and sum the results, multiplying by Δx to approximate the integral. Step 4: For the Trapezoid Rule, calculate the function values at the endpoints of each subinterval. The endpoints are x_i = a + iΔx, where i ranges from 0 to n. Use the formula for the Trapezoid Rule: T ≈ (Δx/2) * [f($$x_0) + 2f$$($$x_1) + 2f$$($$x_2) + ... + 2f(x_(n-1)) + f(x_n$$)]. Step 5: Compare the errors for both methods. The error is the absolute difference between the exact value of the integral, π ln(9/2), and the approximations obtained using the Midpoint Rule and Trapezoid Rule for each value of n (4, 8, 16, and 32). Analyze how the errors decrease as n increases, and observe which method provides a more accurate approximation for each value of n.

54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
59. ∫(from 0 to π) ln(5 + 3cosx) dx = π ln(9/2)

Verified video answer for a similar problem:

Key Concepts

Midpoint Rule

Trapezoid Rule

Error Analysis in Numerical Integration

54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).59. ∫(from 0 to π) ln(5 + 3cosx) dx = π ln(9/2)

Verified video answer for a similar problem:

Key Concepts

Midpoint Rule

Trapezoid Rule

Error Analysis in Numerical Integration

54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
59. ∫(from 0 to π) ln(5 + 3cosx) dx = π ln(9/2)