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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 1 to 2 of 1/s² ds

Verified step by step guidance

1. Identify the function to be integrated: \(f(x) = \frac{1}{x^2}\), and the interval of integration: \([1, 2]\).
2. Recall the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)|\), where \(n\) is the number of subintervals.
3. Compute the second derivative of the function \(f(x)\). Start by finding the first derivative: \(f'(x) = -2 x^{-3}\), then the second derivative: \(f''(x) = 6 x^{-4}\).
4. Determine the maximum value of \(|f''(x)|\) on the interval \([1, 2]\). Since \(f''(x) = 6 / x^4\) is positive and decreasing on \([1, 2]\), the maximum occurs at \(x = 1\), so \(\max |f''(x)| = 6\).
5. Set up the inequality for the error bound to be less than \(10^{-4}\): \(\frac{(2 - 1)^3}{12 n^2} \times 6 < 10^{-4}\). Simplify and solve for \(n\) to find the minimum number of subintervals needed.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the interval into subintervals and approximating the area under the curve as trapezoids. The accuracy depends on the number of subintervals; more subintervals generally yield better approximations.

Error Bound for the Trapezoidal Rule

The error bound for the Trapezoidal Rule estimates the maximum possible error in the approximation. It depends on the second derivative of the function, the length of the interval, and the number of subintervals. Specifically, the error is bounded by (b−a)³/(12n²) times the maximum of |f''(x)| on [a,b].

Second Derivative and Its Role in Error Estimation

The second derivative of the integrand measures the concavity of the function and influences the error in the Trapezoidal Rule. To estimate the error bound, one must find the maximum absolute value of the second derivative on the interval, as it directly affects how closely the trapezoids approximate the curve.