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 x dx

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Identify the integral to approximate: \(\int_{1}^{2} x \, dx\) and note that the interval is from \(a=1\) to \(b=2\).

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

Calculate the \(x\)-values at which the function will be evaluated: \(x_0 = 1\), \(x_1 = 1 + h\), \(x_2 = 1 + 2h\), \(x_3 = 1 + 3h\), and \(x_4 = 2\).

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

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

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