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 -2 to 0 of (x² - 1) dx

Verified step by step guidance

1. Identify the function to be integrated: \(f(x) = x^{2} - 1\) over the interval \([-2, 0]\).
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 absolute value of \(f''(x)\) on the interval \([-2, 0]\): since \(f''(x) = 2\) is constant, \(\max |f''(x)| = 2\).
5. Set the error bound less than the desired tolerance \(10^{-4}\) and solve for \(n\(: \[ \frac{(0 - (-2))^{3}}{12 n^{2}} \times 2 < 10^{-4} \implies \frac{8}{12 n^{2}} \times 2 < 10^{-4} \implies \frac{16}{12 n^{2}} < 10^{-4} \]. Then solve this inequality 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 estimates the maximum possible error in the approximation. It depends on the length of the interval, the number of subintervals, and the maximum value of the second derivative of the function on the interval. This bound helps determine how many subintervals are needed to achieve a desired accuracy.

Second Derivative and Its Role in Error Estimation

The second derivative of the function measures its concavity and affects the error in the Trapezoidal Rule. The maximum absolute value of the second derivative on the interval is used in the error bound formula, indicating how 'curved' the function is and influencing the number of subintervals required for a given error tolerance.