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.
III. Using Simpson's Rule
b. Evaluate the integral directly and find |ES|.
∫ from -2 to 0 of (x² - 1) dx

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First, write down the definite integral to be evaluated: \(\int_{-2}^{0} (x^{2} - 1) \, dx\).

Recall that to evaluate the integral directly, you need to find the antiderivative (indefinite integral) of the integrand \(x^{2} - 1\). The antiderivative of \(x^{2}\) is \(\frac{x^{3}}{3}\), and the antiderivative of \(-1\) is \(-x\).

Combine these results to write the antiderivative function: \(F(x) = \frac{x^{3}}{3} - x\).

Apply the Fundamental Theorem of Calculus by evaluating \(F(x)\) at the upper limit and subtracting the value at the lower limit: calculate \(F(0) - F(-2)\).

To find the absolute error bound \(|E_{S}|\) for Simpson's Rule, recall the error formula: \(|E_{S}| \leq \frac{(b - a)^{5}}{180 n^{4}} \max_{a \leq x \leq b} |f^{(4)}(x)|\). Compute the fourth derivative of \(f(x) = x^{2} - 1\), determine its maximum absolute value on \([-2, 0]\), and substitute all values into the formula.

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Simpson's Rule

Simpson's Rule is a numerical method for approximating definite integrals by dividing the interval into an even number of subintervals and fitting parabolas through the function values. It generally provides more accurate results than the Midpoint or Trapezoidal Rules for smooth functions.

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Exact evaluation involves finding the antiderivative of the integrand and applying the Fundamental Theorem of Calculus to compute the integral's exact value. This provides a benchmark to compare numerical approximations and calculate errors.

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The error bound |ES| estimates the maximum possible difference between the exact integral and the Simpson's Rule approximation. It depends on the fourth derivative of the function and the width of the subintervals, helping assess the accuracy of the numerical method.

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