In collisions between a ball and a striking object (e.g., a baseball bat or tennis racket), the ball changes shape, first compressing and then expanding. If $x$ represents the change in diameter of the ball (e.g., in inches) for $0 \leq x \leq m$ and $f(x)$ represents the force between the ball and striking object (e.g., in pounds), then the area under the curve $y=f(x)$ is proportional to the energy transferred. Suppose that $f_{c}(x)$ is the force during compression and $f_{e}(x)$ is the force during expansion. Explain why $\int_{0}^{m}\left[f_{c}(x)-f_{e}(x)\right] d x$ is proportional to the energy lost by the ball (due to friction) and thus $\int_{0}^{m}\left[f_{c}(x)-f_{e}(x)\right] d x / \int_{0}^{m} f_{c}(x) d x$ is the proportion of energy
lost in the collision. For a baseball and bat, reasonable values are shown (see Adair's book The Physics of Baseball):$$\begin{array}{|c|c|c|c|c|c|}
\hline x \text { (in.) } & 0 & 0.1 & 0.2 & 0.3 & 0.4 \\\hline f_{c}(x)(\mathrm{lb}) & 0 & 250 & 600 & 1200 & 1750 \\\hline f_{e}(x)(\mathrm{lb}) & 0 & 10 & 100 & 270 & 1750 \\\hline\end{array}$$ Use Simpson's Rule to estimate the proportion of energy retained by the baseball.