The measurements of length can be viewed as measuring “top minus bottom” of two functions. The exact answer is found by integrating \(\ds \int_0^{12} \big(f(x)-g(x)\big)\, dx\text{,}\) but of course we don’t know the functions \(f\) and \(g\text{.}\) Our discrete measurements instead allow us to approximate.

We have the following data points:

\begin{equation*}
(0,0),\,(2,2.25),\,(4,5.08),\,(6,6.35),\,(8,5.21),\,(10,2.76),\,(12,0)\text{.}
\end{equation*}

We also have that \(\dx=\frac{b-a}{n} = 2\text{,}\) so Simpson’s Rule gives

\begin{align*}
\text{Area} \amp \approx \frac{2}{3}\Big(1\cdot0+4\cdot2.25+2\cdot5.08+4\cdot6.35+2\cdot5.21+4\cdot2.76+1\cdot0\Big)\\
\amp = 44.01\overline{3} \,\text{units}^2\text{.}
\end{align*}

Since the measurements are in hundreds of feet, square units are given by (100 ft)\(^2 = \)10,000 ft^{2}, giving a total area of 440,133 ft^{2}. (Since we are approximating, we’d likely say the area was about 440,000 ft^{2}, which is a little more than 10 acres.)