In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 2 of sin(x + 1) dx

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Identify the integral to approximate: \(\int_0^2 \sin(x + 1) \, dx\).

Recall the error bound formula for Simpson's Rule: \(|E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq x \leq b} |f^{(4)}(x)|\), where \(n\) is the number of subintervals (and must be even).

Find the fourth derivative of the integrand \(f(x) = \sin(x + 1)\). Since \(f(x) = \sin(x + 1)\), its derivatives cycle every four steps: \(f^{(1)}(x) = \cos(x + 1)\), \(f^{(2)}(x) = -\sin(x + 1)\), \(f^{(3)}(x) = -\cos(x + 1)\), and \(f^{(4)}(x) = \sin(x + 1)\).

Determine the maximum value of \(|f^{(4)}(x)|\) on the interval \([0, 2]\). Since \(f^{(4)}(x) = \sin(x + 1)\), find the maximum absolute value of \(\sin(x + 1)\) for \(x\) in \([0, 2]\).

Set up the inequality for the error bound to be less than \(10^{-4}\): \(\frac{(2 - 0)^5}{180 n^4} \max |f^{(4)}(x)| < 10^{-4}\). Solve this inequality for \(n\) to find the minimum number of subintervals needed.

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주요 개념

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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 quadratic polynomials to the function. It generally provides more accurate results than the trapezoidal rule for smooth functions.

Error Bound for Simpson's Rule

The error bound for Simpson's Rule depends on the fourth derivative of the integrand. Specifically, the error magnitude is less than (K(b - a)^5) / (180 n^4), where K is the maximum absolute value of the fourth derivative on [a, b], and n is the number of subintervals.

Estimating the Number of Subintervals

To ensure the approximation error is below a desired threshold, solve the error bound inequality for n, the number of subintervals. This involves finding the maximum of the fourth derivative, substituting into the error formula, and calculating the minimum n that satisfies the error requirement.

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)∫ from 0 to 2 of sin(x + 1) dx