Single Variable Calculus, Chapter 6, 6.4, Section 6.4, Problem 6

Use the Midpoint Rule to estimate the work done by the force $f(x)$ (in newtons) in moving an object $x$ meters from $x = 4$ to $x = 20$.

$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline\\ x & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\ \hline\\ f(x) & 5 & 5.8 & 7.0 & 8.8 & 9.6 & 8.2 & 6.7 & 5.2 & 4.1\\ \hline \end{array}$

Since the range is separated into 4 segments in Midpoint, we have..


$\begin{equation} \begin{aligned} W = \int^{20}_4 f(x) dx \approx & \Delta x [f(6) + f(10) + f(14) + f(18)] \\ \\ \approx & \frac{20 - 4}{4} [5.8 + 8.8 + 8.2 + 5.2] \\ \\ \approx & 112 \text{ Joules} \end{aligned} \end{equation}$