Question

49–52. {Use of Tech} Simpson’s RuleApply 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)

Accepted Answer

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: $$ S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n) ight] $$, where $$ \Delta x = \frac{b-a}{n} . $$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. 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. 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. 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 $$	ext{Error} = |S_n - 	ext{Exact Value}|.$$

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Key Concepts

Simpson's Rule

Trapezoidal Rule

Error Analysis