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
a. Estimate the integral with n = 4 steps and find an upper bound for |ES|.
∫ from 0 to 2 of (t³ + t) dt

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Identify the function to integrate: \(f(t) = t^{3} + t\), and the interval of integration: \([0, 2]\).

Determine the step size \(h\) using the formula \(h = \frac{b - a}{n}\), where \(a = 0\), \(b = 2\), and \(n = 4\).

Calculate the values of the function \(f(t)\) at the equally spaced points \(t_0, t_1, t_2, t_3, t_4\) where \(t_i = a + i \cdot h\) for \(i = 0, 1, 2, 3, 4\).

Apply Simpson's Rule formula for \(n=4\) steps: \(S = \frac{h}{3} \left[ f(t_0) + 4f(t_1) + 2f(t_2) + 4f(t_3) + f(t_4) \right]\). Substitute the function values calculated in the previous step into this formula.

To find an upper bound for the error \(|E_S|\), use the error bound formula for Simpson's Rule: \(|E_S| \leq \frac{(b - a)^5}{180 n^4} \max_{a \leq t \leq b} |f^{(4)}(t)|\). Calculate the fourth derivative \(f^{(4)}(t)\) of the function, find its maximum absolute value on \([0, 2]\), and substitute all values into the error bound 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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The error bound for Simpson's Rule estimates the maximum possible difference between the exact integral and the approximation. It depends on the fourth derivative of the function and the number of subintervals, providing a way to assess the accuracy of the approximation.

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Integrating polynomial functions like t³ + t over an interval involves finding the exact area under the curve. Understanding the integral's exact value helps compare and validate numerical approximations such as Simpson's Rule.

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