49–52. {Use of Tech} Simpson’s Rule
Apply Simpson’s Rule to the following integrals. It is easiest to obtain the Simpson’s Rule approximations from the Trapezoid Rule approximations, as in Example 8. Make a table similar to Table 8.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
51. ∫(from 0 to π) e⁻ᵗ sin(t) dt = ½(e⁻ᵖⁱ + 1)

Verified step by step guidance

1. Step 1: Understand Simpson's Rule. Simpson's Rule is a numerical method for approximating definite integrals. It uses parabolic arcs to approximate the curve of the function. The formula for Simpson's Rule is: \( S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n) \right] \), where \( \Delta x = \frac{b-a}{n} \).
2. Step 2: Divide the interval \([0, \pi]\) into \(n\) subintervals. For \(n = 4, 8, 16, 32\), calculate \(\Delta x = \frac{\pi}{n}\). These subintervals will be used to evaluate the function \(e^{-t} \sin(t)\) at specific points.
3. Step 3: Compute the function values \(f(x_i) = e^{-x_i} \sin(x_i)\) at the endpoints \(x_0 = 0\) and \(x_n = \pi\), as well as at the intermediate points \(x_1, x_2, \dots, x_{n-1}\). Use these values in the Simpson's Rule formula.
4. Step 4: Use the Trapezoid Rule approximations to simplify the computation of Simpson's Rule. Recall that Simpson's Rule can be expressed as \(S_n = \frac{4T_{2n} - T_n}{3}\), where \(T_n\) and \(T_{2n}\) are the Trapezoid Rule approximations for \(n\) and \(2n\) subintervals, respectively.
5. Step 5: Create a table similar to Table 8.8. For each \(n = 4, 8, 16, 32\), calculate the Simpson's Rule approximation \(S_n\), compare it to the exact value \(\frac{1}{2}(e^{-\pi} + 1)\), and compute the error \(\text{Error} = |S_n - \text{Exact Value}|\).

Key Concepts

Simpson's Rule

Simpson's Rule is a numerical method for approximating the definite integral of a function. It uses parabolic segments to estimate the area under a curve, providing a more accurate approximation than the Trapezoidal Rule, especially for functions that are smooth. The formula involves evaluating the function at equally spaced intervals and applying a specific weighting to these values.

Trapezoidal Rule

The Trapezoidal Rule is another numerical integration technique that approximates the area under a curve by dividing it into trapezoids rather than rectangles. The area of each trapezoid is calculated and summed to provide an estimate of the total integral. This method is simpler but generally less accurate than Simpson's Rule, particularly for functions with curvature.

Error Analysis

Error analysis in numerical integration involves assessing the difference between the exact value of an integral and its numerical approximation. This is crucial for understanding the accuracy of methods like Simpson's Rule and the Trapezoidal Rule. By calculating the error for different numbers of intervals (n), one can evaluate how the approximation improves with increased precision and identify the effectiveness of the chosen method.