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).
54. ∫(from 0 to π/2) sin⁶x dx = 5π/32

Verified step by step guidance

1. Identify the integral to approximate: \(\int_0^{\frac{\pi}{2}} \sin^6 x \, dx\) with exact value \(\frac{5\pi}{32}\).
2. Recall the formulas for the Midpoint Rule and the Trapezoid Rule for approximating integrals with \(n\) subintervals: - Midpoint Rule: \(M_n = \Delta x \sum_{i=1}^n f\left(x_{i-\frac{1}{2}}\right)\), - Trapezoid Rule: \(T_n = \frac{\Delta x}{2} \left(f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right)\), where \(\Delta x = \frac{b - a}{n}\) and \(x_i = a + i \Delta x\).
3. For each \(n = 4, 8, 16, 32\), calculate \(\Delta x = \frac{\pi/2 - 0}{n} = \frac{\pi}{2n}\) and determine the sample points: - For Midpoint Rule, use midpoints \(x_{i-\frac{1}{2}} = a + \left(i - \frac{1}{2}\right) \Delta x\), - For Trapezoid Rule, use endpoints \(x_i = a + i \Delta x\) for \(i = 0, 1, ..., n\).
4. Evaluate \(f(x) = \sin^6 x\) at the required points for each rule and sum according to the formulas to get approximations \(M_n\) and \(T_n\).
5. Calculate the absolute errors for each \(n\) and each rule by subtracting the approximate values from the exact value \(\frac{5\pi}{32}\): \(\text{Error}_{Midpoint} = \left| M_n - \frac{5\pi}{32} \right|\), \(\text{Error}_{Trapezoid} = \left| T_n - \frac{5\pi}{32} \right|\). Compare these errors to analyze which rule gives better accuracy as \(n\) increases.

Key Concepts

Numerical Integration Methods

Numerical integration approximates definite integrals when exact solutions are difficult. The Midpoint Rule estimates the integral using function values at subinterval midpoints, while the Trapezoid Rule uses linear approximations between endpoints. Both methods divide the interval into subintervals to improve accuracy.

Error Analysis in Numerical Integration

Error analysis evaluates how close a numerical approximation is to the exact integral. The Midpoint Rule error depends on the second derivative of the function and generally decreases faster than the Trapezoid Rule error, which also depends on the second derivative but with a different constant. Understanding error behavior helps compare method efficiency.

Effect of Subinterval Number (n) on Accuracy

Increasing the number of subintervals (n) in numerical methods typically reduces approximation error. Both Midpoint and Trapezoid Rules improve as n grows, but their error rates differ. Analyzing errors for n = 4, 8, 16, and 32 shows how refinement impacts accuracy and convergence speed.