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

Accepted Answer

Identify the integral to approximate: $$\$$int_0$$^{\frac{\pi}{2}} \sin^6$ x \, dx$$ with exact value $$\frac{5\pi}{32}. $$Recall the formulas for the Midpoint Rule and the Trapezoid Rule for approximating integrals with $$n$$ subintervals:
- Midpoint Rule: $$M_n = \Delta x \sum_{i=1}^n f\left(x_{i-\frac{1}{2}}ight)$$,
- Trapezoid Rule: $$T_n = \frac{\Delta x}{2} \left(f(x_0$) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)ight)$$,
where $$\Delta x = \frac{b - a}{n}$$ and $$x_i = a + i \Delta x. $$For each $$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}ight) \Delta x$$,
- For Trapezoid Rule, use endpoints $$x_i = a + i \Delta x$$ for $$i = 0, 1, ..., n. $$Evaluate $$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. $$Calculate the absolute errors for each $$n$$ and each rule by subtracting the approximate values from the exact value $$\frac{5\pi}{32}$$:
$$	ext{Error}_{Midpoint} = \left| M_n - \frac{5\pi}{32} ight|$$,
$$	ext{Error}_{Trapezoid} = \left| T_n - \frac{5\pi}{32} ight|.
$$Compare these errors to analyze which rule gives better accuracy as $$n$$ increases.

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

Risposta video verificata per un problema simile:

Concetti chiave

Numerical Integration Methods

Error Analysis in Numerical Integration

Effect of Subinterval Number (n) on Accuracy

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

Risposta video verificata per un problema simile:

Concetti chiave

Numerical Integration Methods

Error Analysis in Numerical Integration

Effect of Subinterval Number (n) on Accuracy

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