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 -1 to 1 of (x² + 1) dx

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Identify the function to be integrated: \(f(x) = x^2 + 1\) over the interval \([-1, 1]\).

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).

Compute the fourth derivative of the function \(f(x)\). Since \(f(x) = x^2 + 1\), find \(f^{(4)}(x)\).

Determine the maximum value of \(|f^{(4)}(x)|\) on the interval \([-1, 1]\).

Set the error bound \(|E_S|\) less than \(10^{-4}\) and solve the inequality for \(n\) to find the minimum number of subintervals needed.

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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 parabolas through the function values. It generally provides more accurate results than the trapezoidal or midpoint rules for smooth functions.

Error Bound for Simpson's Rule

The error bound for Simpson's Rule depends on the fourth derivative of the integrand and the number of subintervals. Specifically, the error magnitude is less than (K(b−a)^5)/(180n^4), where K bounds the 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 on the interval and rearranging the error formula to find the minimum n that satisfies the error tolerance.

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 -1 to 1 of (x² + 1) dx