Question

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 2 to 4 of 1/(s - 1)² ds

Accepted Answer

Identify the function to be integrated: $$f(s) = \frac{1}{(s - 1)^2$}$$, and the interval of integration is from $$a = 2 to$$ $$b = 4. $$Recall the error bound formula for Simpson's Rule: the error $$E_S$$ satisfies

$$|E_S| \leq \frac{(b - a)^5$}{180 n^4$} \max_{a \leq s \leq b} |f$$^{(4)}$$(s)|$$,

where $$n is $$the number of subintervals (which must be even), and $$f^{(4)}(s$$)$$ is the fourth derivative of f$. Compute the fourth derivative f$$^{(4)}$$(s) of $$the function $$f(s) = (s - 1$$)^{-2}$$. This involves differentiating f(s$$)$$ four times with respect to s$. Determine the maximum value of $$|$$f^{(4)}(s$$)|$$ on the interval $$[2, 4]$$. Since f$$^{(4)}$$(s) is $$continuous on this interval, evaluate it at critical points and endpoints to find the maximum. Set the error bound less than the desired tolerance $$10^{-4}$$ and solve for $$n$$:

$$\frac{(4 - 2)^5$}{180 n^4$} \max |f$$^{(4)}$$(s)|  \left( \frac{(4 - 2)^5$ \max |f$$^{(4)}$$(s)|}{180 	imes 10$$^{-4}$$} \$$right)^{1/4}$$.

Choose the smallest even integer n$ that satisfies this inequality.

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 2 to 4 of 1/(s - 1)² ds

Verified video answer for a similar problem:

Key Concepts

Simpson's Rule

Error Bound for Simpson's Rule

Fourth Derivative of the Integrand

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 2 to 4 of 1/(s - 1)² ds

Verified video answer for a similar problem:

Key Concepts

Simpson's Rule

Error Bound for Simpson's Rule

Fourth Derivative of the Integrand

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 2 to 4 of 1/(s - 1)² ds