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 1 of (x² + 1) dx

Verified step by step guidance

1. Identify the function to be integrated: \(f(x) = x^{2} + 1\) over the interval \([-1, 1]\).
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: \(f''(x) = \frac{d^{2}}{dx^{2}}(x^{2} + 1) = 2\).
4. Determine the maximum value of \(|f''(x)|\) on the interval \([-1, 1]\), which is \(2\) since \(f''(x)\) is constant.
5. Set up the inequality for the error bound to be less than \(10^{-4}\) and solve for \(n\): \(\frac{(1 - (-1))^{3}}{12 n^{2}} \times 2 < 10^{-4}\). 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 sum of these trapezoidal areas estimates the integral, with accuracy improving as the number of subintervals increases.

Error Bound for the Trapezoidal Rule

The error bound for the Trapezoidal Rule depends on the second derivative of the function being integrated. Specifically, the error magnitude is at most (K(b - a)^3) / (12n^2), where K is the maximum absolute value of the second derivative on [a, b], and n is the number of subintervals. This formula helps estimate how many subintervals are needed for a desired accuracy.

Second Derivative and Its Role in Error Estimation

The second derivative of the integrand measures the function's concavity and affects the error in the Trapezoidal Rule. A larger maximum second derivative on the interval implies a larger potential error, requiring more subintervals to achieve a given accuracy. Calculating or bounding this derivative is essential for applying the error formula.