The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
III. Using Simpson's Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.
∫ from 0 to π of sin(t) dt

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Identify the integral to approximate: \(\int_0^{\pi} \sin(t) \, dt\) and note that \(n=4\) subintervals will be used for Simpson's Rule.

Calculate the step size \(h\) using the formula \(h = \frac{b - a}{n} = \frac{\pi - 0}{4} = \frac{\pi}{4}\).

Determine the partition points: \(t_0 = 0\), \(t_1 = \frac{\pi}{4}\), \(t_2 = \frac{\pi}{2}\), \(t_3 = \frac{3\pi}{4}\), and \(t_4 = \pi\).

Apply Simpson's Rule formula: \(S_n = \frac{h}{3} \left[f(t_0) + 4f(t_1) + 2f(t_2) + 4f(t_3) + f(t_4)\right]\), where \(f(t) = \sin(t)\).

To find the error bound \(|E_S|\), use the formula for Simpson's Rule error bound: \(|E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq t \leq b} |f^{(4)}(t)|\). Calculate the fourth derivative of \(f(t) = \sin(t)\), find its maximum absolute value on \([0, \pi]\), and substitute all values to estimate the error bound.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Simpson's Rule

Simpson's Rule is a numerical method for approximating definite integrals by dividing the interval into an even number of subintervals and fitting parabolas through the function values. It generally provides more accurate results than the Midpoint or Trapezoidal Rules for smooth functions. The formula combines function values at equally spaced points with specific weights to estimate the integral.

Error Bound for Simpson's Rule

The error bound for Simpson's Rule estimates the maximum possible difference between the true integral and the approximation. It depends on the fourth derivative of the integrand, the interval length, and the number of subintervals. Specifically, the error bound helps assess the accuracy of the approximation and ensures the estimate is within a known tolerance.

Definite Integral of sin(t) from 0 to π

The integral of sin(t) from 0 to π represents the area under one half-wave of the sine curve, which equals 2 exactly. Understanding this integral provides a benchmark to compare numerical approximations. Since sin(t) is smooth and well-behaved on [0, π], it is suitable for applying Simpson's Rule and analyzing error bounds.

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.III. Using Simpson's Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ES|.∫ from 0 to π of sin(t) dt