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 2 of x dx

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

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 (must be even), and \( f^{(4)}(x) \) is the fourth derivative of \( f(x) \).

Compute the fourth derivative of \( f(x) = x \). Since \( f(x) = x \), \( f'(x) = 1 \), \( f''(x) = 0 \), and all higher derivatives including \( f^{(4)}(x) = 0 \).

Since \( f^{(4)}(x) = 0 \), the error bound formula implies the error is zero for any \( n \), meaning Simpson's Rule will give the exact value for this integral regardless of the number of subintervals.

Therefore, the minimum number of subintervals needed to achieve an error less than \( 10^{-4} \) is the smallest even number, which is \( n = 2 \), but in fact, any \( n \) will suffice.

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Simpson's Rule

Simpson's Rule is a numerical method for approximating definite integrals by fitting parabolas through subintervals of the function. It uses an even number of subintervals and 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 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 1 to 2 of x dx