4. $[-/ 1$ Points] DETAILS SCALC9 4.2.009. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Use the Midpoint Rule with $n=4$ to approximate the integral. \[ \int_{6}^{14} x^{2} d x \] \[ M_{4}= \] Need Help? Read It Submit Answer
Solution
Step 1 ：Define the given values: \(a = 6\), \(b = 14\), \(n = 4\), and \(f(x) = x^2\).
Step 2 ：Calculate \(\Delta x\) using the formula \(\Delta x = \frac{b - a}{n}\). Substituting the given values, we get \(\Delta x = 2.0\).
Step 3 ：Find the midpoints of each subinterval using the formula \(x_i^* = a + \frac{\Delta x}{2} + (i - 1) \Delta x\). The midpoints are \([ 7. 9. 11. 13.]\).
Step 4 ：Calculate the sum of the function values at the midpoints. Substituting \(x^2\) into the midpoints, we get \(sum_f_midpoints = 420.0\).
Step 5 ：Finally, calculate \(M_4\) using the Midpoint Rule formula \(M_n = \Delta x \sum_{i=1}^{n} f(x_i^*)\). Substituting the calculated values, we get \(M_4 = 840.0\).
Step 6 ：The approximation of the integral using the Midpoint Rule with \(n = 4\) is \(\boxed{840}\).
