Chapter 6.1, Problem 4E

To determine

To find: the area of the shaded region.

Answer to Problem 4E

The area of the shaded region is 9.

Explanation of Solution

Given information: Given image is shown below.

  

Calculation:

The left and right boundary curves of the shaded region are:

  $x_L=y^2-4y\text{and}{x}_R=2y-y^2.$

So a typical rectangle will have a length of,

  $x_R-x_L=2y-y^2-(y^2-4y)=6y-2y^2.$

To find the area of the shaded region, integrate between y −values, y = 0 and y = 3.

  $\begin{array}{c}{\displaystyle \underset{0}{\overset{3}{\int }}(6y-2y^2)}dy=\left[3y^2-\frac{2y^3}{3}\right]\begin{array}{c}3\\ 0\end{array}.\\ $ =(3{(3)}^2-\frac{2{(3)}^3}{3})-(3{(0)}^2-\frac{2{(0)}^3}{3}).$=9-0\\ =9\end{array}$

The area of the shaded region is 9.