Chapter 6.1, Problem 1E

To determine

To find: the area of the shaded region.

Answer to Problem 1E

The area of the shaded region is 10.6667.

Explanation of Solution

Given information: Given image is shown below.

  

Calculation:

The top and bottom functions of the shaded region are:

  $y_T=5x-x^2\text{and}{y}_B=x.$

So a typical rectangle will have a height of,

  $y_T-y_B=(5x-x^2)-x=4x-x^2.$

To find the area of the shaded region, integrate between x −values, x = 0 and x = 4.

  $\begin{array}{c}{\displaystyle \underset{0}{\overset{4}{\int }}(4x-x^2})dx={\displaystyle \underset{0}{\overset{4}{\int }}4xdx}-{\displaystyle \underset{0}{\overset{4}{\int }}{x}^{2}dx}.\\ =4{\displaystyle \underset{0}{\overset{4}{\int }}xdx}-{\displaystyle \underset{0}{\overset{4}{\int }}{x}^{2}dx}.\\ =4\frac{x^2}{2}|\begin{array}{c}4\\ 0\end{array}-\frac{x^3}{3}|\begin{array}{c}4\\ 0\end{array}.\\ =(4.\frac{4^2}{2}-4.\frac{0^2}{2})-(\frac{4^3}{3}-\frac{0^3}{3}).\\ =32-\frac{64}{3}.\\ =10.6667\end{array}$

The area of the shaded region is 10.6667.