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 1 to 2 of 1 / s² ds

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Identify the function to integrate: \(f(s) = \frac{1}{s^2}\), and the interval of integration: \([1, 2]\).

Calculate the step size \(h\) using the formula \(h = \frac{b - a}{n}\), where \(a = 1\), \(b = 2\), and \(n = 4\).

Determine the partition points: \(s_0 = 1\), \(s_1 = 1 + h\), \(s_2 = 1 + 2h\), \(s_3 = 1 + 3h\), and \(s_4 = 2\).

Apply the Trapezoidal Rule formula: \(T_n = \frac{h}{2} \left[f(s_0) + 2f(s_1) + 2f(s_2) + 2f(s_3) + f(s_4)\right]\), where you substitute the function values at each partition point.

To find the upper bound for the error \(|E_T|\), use the error bound formula for the Trapezoidal Rule: \(|E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq s \leq b} |f''(s)|\). Calculate the second derivative \(f''(s)\), find its maximum absolute value on \([1, 2]\), and substitute all values to estimate the error bound.

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