Question

the widths (in meters) of a kidney - shaped swimming pool were measured at 2 - meter intervals as indicated in the figure.use the midpoint rule with ( n = 4 ) to estimate the area (in ( m^{2} )) of the pool.

Explanation:

Step1: Determine the width of each sub - interval

The interval length \(a = 0\), \(b=8\) (since \(n = 4\) and \(\Delta x=\frac{b - a}{n}\), and the measurement interval is \(2\) meters, total length \(L=8\) meters). \(\Delta x=\frac{8 - 0}{4}=2\) meters.

Step2: Identify the mid - points

The mid - points of the four sub - intervals \([0,2]\), \([2,4]\), \([4,6]\), \([6,8]\) are \(x_1 = 1\), \(x_2 = 3\), \(x_3 = 5\), \(x_4 = 7\). The corresponding widths (function values) are \(y_1=10.8\), \(y_2 = 8.4\), \(y_3=7.2\), \(y_4=7.2\)

Step3: Apply the mid - point rule formula

The mid - point rule formula for \(\int_{a}^{b}f(x)dx\approx\Delta x\sum_{i = 1}^{n}f(x_i)\). Here, the area \(A\approx\Delta x(y_1 + y_2+y_3 + y_4)\)
Substitute \(\Delta x = 2\), \(y_1 = 10.8\), \(y_2 = 8.4\), \(y_3=7.2\), \(y_4=7.2\) into the formula:
\(A\approx2\times(10.8 + 8.4+7.2 + 7.2)\)
First, calculate the sum inside the parentheses: \(10.8+8.4 + 7.2+7.2=(10.8+8.4)+(7.2 + 7.2)=19.2+14.4 = 33.6\)
Then, \(A\approx2\times33.6\)

Answer:

\(67.2\space m^{2}\)