Chapter 7.2, Problem 45E

To determine

To estimate: It needs to be determined the area of the lot using trapezoidal sum rule for the table shown in the textbook.

Answer to Problem 45E

The area of the lot is 8180 square feet.

Explanation of Solution

Given information: A table is shown in the textbook where the lot is 100 feet across from west to east but the boundaries on north and south sides are curved and for estimating the area north south measurements are taken of the lot every 10 feet from western boundary.

Formula used: The Trapezoid rule is used which states that when $\left[a,b\right]$ is divided into n subintervals each length $\Delta x$ with endpoints at $\left\{x_0,x_1,x_2,\mathrm{...},x_n\right\}$ then one has as:

  $T_n=\frac{\Delta x}{2}\left[f\left(x_0\right)+2f\left(x_1\right)+2f\left(x_2\right)+\mathrm{...}+2f\left(x_{n-2}\right)+2f\left(x_{n-1}\right)+f\left(x_n\right)\right]$

Calculation: The north south distance can be described by the function as $f\left(x\right)$ , so the area of the lot is given by $A={\displaystyle \underset{0}{\overset{100}{\int }}f\left(x\right)dx}$ .

As the width of each interval is as $\Delta x=10$, thus using trapezoid rule one gets as:

  $\begin{array}{c}A=\frac{\Delta x}{2}\left[\begin{array}{l}f\left(0\right)+2f\left(10\right)+2f\left(20\right)+2f\left(30\right)+2f\left(40\right)+2f\left(50\right)\\ +2f\left(60\right)+2f\left(70\right)+2f\left(80\right)+2f\left(90\right)+f\left(100\right)\end{array}\right]\\ =\frac{10}{2}\left[75+2\left(74\right)+2\left(78\right)+2\left(82\right)+2\left(85\right)+2\left(88\right)+2\left(90\right)+2\left(85\right)+2\left(82\right)+2\left(79\right)+\left(75\right)\right]\\ =8180\end{array}$

Hence the area of the lot is 8180 square feet.