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 step by step guidance

1. 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.
2. 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.
3. 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.
4. 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)].
5. 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.

Key Concepts

Midpoint Rule

The Midpoint Rule is a numerical method for approximating the definite integral of a function. It involves dividing the interval into subintervals, calculating the function's value at the midpoint of each subinterval, and then summing these values multiplied by the width of the subintervals. This method tends to provide a better approximation for functions that are relatively smooth and continuous.

Trapezoid Rule

The Trapezoid Rule is another numerical integration technique that approximates the area under a curve by dividing the interval into subintervals and forming trapezoids. The area of each trapezoid is calculated using the average of the function values at the endpoints of each subinterval. This method is generally more accurate than the Midpoint Rule for functions that are linear or nearly linear over the subintervals.

Error Analysis in Numerical Integration

Error analysis in numerical integration involves assessing the difference between the exact value of an integral and its numerical approximation. The error can depend on the method used, the number of subintervals, and the behavior of the function being integrated. Understanding how the error decreases with increasing subintervals (n) is crucial for comparing the effectiveness of different numerical methods like the Midpoint and Trapezoid Rules.