Chapter 8.2, Problem 3E

To determine

To Find:

To find the area of the given shaded region analytically.

Answer to Problem 3E

The area of the shaded region is $\frac{1}{12}$

Explanation of Solution

Given information:

The given function is $x=y^3$ and $x=y^2$ in the interval $0\le y\le 1$ .

  

If a region is bounded by a right curve $x_R=g_{L\left(y\right)}$ and a left curve $x_L=g_{R\left(y\right)}$ from $y=c$ to $y=d$ , then the area of this region is

  $A={\displaystyle \underset{c}{\overset{d}{\int }}\left(x_R-x_L\right)}dy$

  $A={\displaystyle \underset{0}{\overset{1}{\int }}\left(y^2-y^3\right)dy}$

  $\begin{array}{l}={\left[\frac{y^3}{3}-\frac{y^4}{4}\right]}_0^1\\ =\frac{1}{3}-\frac{1}{4}\\ =\frac{1}{12}\end{array}$

To graph:

  

Here for $0\le y\le 1$ , $x=y^2$ is the blue curve, which is on the right and $x=y^3$ is the red curve, which is on the left.

Therefore, the area of the shaded region is $\frac{1}{12}$ .