Chapter 11.5, Problem 39E

To determine

To calculate:

The area of a property.

Answer to Problem 39E

The area is $105208.33$ square feet.

Explanation of Solution

Given Information:

  $y=\left(-3.0\times {10}^{-6}\right)x^3+0.002x^2-1.05x+400$

Calculation:

Consider the following equation

  $y=\left(-3.0\times {10}^{-6}\right)x^3+0.002x^2-1.05x+400$

Graph the equation the graphical utility,

  

The above graph shows the equation $y=\left(-3.0\times {10}^{-6}\right)x^3+0.002x^2-1.05x+400$ .

From the graph, the intervals are $\left[0,500\right]$

Now use the formula $A=\underset{x\to \infty }{\lim }\left({\displaystyle \sum _{i=1}^{n}f}\left(a+\frac{\left(b-a\right)i}{n}\right)\left(\frac{b-a}{n}\right)\right)$ to get the exact area.

Therefore,

  $\begin{array}{l}A=\underset{x\to \infty }{\lim }\left({\displaystyle \sum _{i=1}^{n}f}\left(a+\frac{\left(b-a\right)i}{n}\right)\left(\frac{b-a}{n}\right)\right)\\ =\underset{x\to \infty }{\lim }\left({\displaystyle \sum _{i=1}^n\left(\left(-3.0\times {10}^{-6}\right){\left(\frac{500i}{n}\right)}^3+0.002{\left(\frac{500i}{n}\right)}^2-1.05\left(\frac{500i}{n}\right)+400\right)}\left(\frac{500i}{n}\right)\right)\\ =\underset{x\to \infty }{\lim }\left({\displaystyle \sum _{i=1}^n\left(-\frac{375i^3}{n^3}+{\frac{500i}{n^2}}^2-\frac{525i}{n}+400\right)}\frac{500}{n}\right)\end{array}$

Use the following summation formula,

  $\begin{array}{l}{\displaystyle \sum _{i=1}^n{i}^3}=\frac{n^2{\left(n+1\right)}^2}{4}\\ {\displaystyle \sum _{i=1}^n{i}^2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}\\ {\displaystyle \sum _{i=1}^{n}i=}\frac{n\left(n+1\right)}{2}\end{array}$

Hence,

  $\begin{array}{l}A=\underset{x\to \infty }{\lim }\left(-\left(\frac{187500}{n^4}\cdot \frac{n^2{\left(n+1\right)}^2}{4}\right)+\left(\frac{250000}{n^3}\cdot \frac{n\left(n+1\right)\left(2n+1\right)}{6}\right)-\left(\frac{262500}{n^2}\cdot \frac{n\left(n+1\right)}{4}\right)+200000\right)\\ =\underset{x\to \infty }{\lim }\left(-46875.\frac{{\left(n+1\right)}^2}{n^2}+\left(\frac{125000}{3}.\frac{2n^2+3n+1}{n^2}\right)-\left(131250.\frac{n^2+n}{n^2}\right)+200000\right)\\ =-46875+\frac{250000}{3}-131250+200000\\ =105208.33\end{array}$

Hence, the area is $105208.33$ square feet.