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 3 of 1/√(x + 1) dx

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Identify the integral to approximate: \(\int_0^3 \frac{1}{\sqrt{x + 1}} \, dx\).

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).

Compute the fourth derivative \(f^{(4)}(x)\) of the function \(f(x) = \frac{1}{\sqrt{x + 1}} = (x + 1)^{-1/2}\).

Find the maximum absolute value of \(f^{(4)}(x)\) on the interval \([0, 3]\) by evaluating or estimating \(|f^{(4)}(x)|\) over this range.

Set the error bound \(\frac{(3 - 0)^5}{180 n^4} \max |f^{(4)}(x)| < 10^{-4}\) and solve this inequality for \(n\) to find the minimum even 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 quadratic polynomials to the function. 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) / (180 n^4), where K bounds the absolute value of the fourth derivative on [a, b], and n is the number of subintervals.

Estimating the Fourth Derivative and Choosing n

To estimate the minimum number of subintervals n, one must find an upper bound K for the absolute value of the fourth derivative of the integrand on the interval. Then, solve the error inequality for n to ensure the error is below the desired tolerance, such as 10^-4.

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 3 of 1/√(x + 1) dx