The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.
II. Using the Trapezoidal Rule
a. Estimate the integral with n = 4 steps and find an upper bound for |ET|.
∫ from -2 to 0 of (x² - 1) dx

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Identify the integral to approximate: \(\int_{-2}^{0} (x^{2} - 1) \, dx\) and note that we will use the Trapezoidal Rule with \(n = 4\) subintervals.

Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n}\), where \(a = -2\) and \(b = 0\).

Determine the \(x\)-values at the endpoints of each subinterval: \(x_0 = -2\), \(x_1 = -2 + \Delta x\), \(x_2 = -2 + 2\Delta x\), \(x_3 = -2 + 3\Delta x\), and \(x_4 = 0\).

Evaluate the function \(f(x) = x^{2} - 1\) at each of these points to get \(f(x_0), f(x_1), f(x_2), f(x_3), f(x_4)\).

Apply the Trapezoidal Rule formula: \(T_n = \frac{\Delta x}{2} \left[f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)\right]\) to estimate the integral.

To find an upper bound for the error \(|E_T|\), use the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)|\).

Compute the second derivative \(f''(x)\) of the function \(f(x) = x^{2} - 1\), then find its maximum absolute value on the interval \([-2, 0]\).

Substitute the values of \(a\), \(b\), \(n\), and \(\max |f''(x)|\) into the error bound formula to get the upper bound for \(|E_T|\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the interval into n subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoidal areas provides an estimate of the integral, improving accuracy as n increases.

Error Bound for the Trapezoidal Rule

The error bound |ET| for the Trapezoidal Rule estimates the maximum possible difference between the true integral and its approximation. It depends on the second derivative of the function, the interval length, and the number of subintervals, providing a way to gauge the accuracy of the approximation.

Definite Integral of a Polynomial Function

A definite integral of a polynomial function, like ∫(x² - 1) dx, represents the net area between the curve and the x-axis over a specified interval. Understanding how to integrate polynomials exactly helps in comparing numerical approximations to the true value.

The instructions for the integrals in Exercises 1–10 have three parts, one for the Midpoint Rule, one for the Trapezoidal Rule, and one for Simpson’s Rule.II. Using the Trapezoidal Rulea. Estimate the integral with n = 4 steps and find an upper bound for |ET|.∫ from -2 to 0 of (x² - 1) dx