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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