Question

The widths (in meters) of a kidney-shaped swimming pool were measured at 2 -meter intervals as indicated in the figure. Use Simpson's Rule to estimate the area of the pool.

Short answer

The area of the swimming pool is \(84\;{{\rm{m}}^2}\).

Step-by-step solution

The general expression for Simpson’s rule 

Simpson's rule:

\(\int_a^b f (x)dx \approx \frac{{\Delta t}}{3}\left( {{y_0} + 4{y_1} + 2{y_2} + 4{y_3} + 2{y_4} + \cdots + 4{y_{n - 1}} + {y_n}} \right)\)

where\(\Delta t = \frac{{b - a}}{n}\)

Area of the swimming pool 

The widths of the swimming pool were measured at 2 -meter intervals.

From the given figure, observe that the number of subintervals is 8 and so \(x\) is continuous from 0 to 16.

The width of the subinterval \((\Delta t)\) is calculated as follows.

\(\begin{aligned}{l}\Delta t = \frac{{16 - 0}}{8}\\ = 2\end{aligned}\)

Evaluate the area of the pool as follows.

\(\begin{aligned}{l}\int_0^{16} f (x)dx\\ \approx \frac{2}{3}(0 + 4(6.2) + 2(7.2) + 4(6.8) + 2(5.6) + 4(5.0) + 2(4.8) + 4(4.8) + 0)\\ \approx \frac{2}{3}(24.8 + 14.4 + 27.2 + 11.2 + 20.0 + 9.6 + 19.2)\\ \approx \frac{2}{3}(126.4)\end{aligned}\)

Solve the equation further

\(\begin{aligned}{l} \approx 84.27\\ \approx 84\end{aligned}\)

Therefore, the area of the swimming pool is \(84\;{{\rm{m}}^2}\).