Chapter 8.3, Problem 24E

To determine

To find : the volume of the solid.

Answer to Problem 24E

The volume of the solid is $4\pi$ .

Explanation of Solution

Given information : The region enclosed about the y -axis $x=y^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},x=0,y=2$ .

Calculation :

Area of each cross-sectional disk:

  $\begin{array}{l}\Rightarrow \pi {\left(radius\right)}^2\\ \Rightarrow \pi {\left(y^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-0\right)}^2\\ \Rightarrow \pi y^3\end{array}$

Volume of the solid obtained after rotating the given region about y -axis is:

  $\begin{array}{l}V={\displaystyle \int \pi y^3}\\ V={\displaystyle \underset{0}{\overset{2}{\int }}\pi y^{3}dy}\\ V=\pi \left[{\displaystyle \underset{0}{\overset{2}{\int }}\left(y^3\right)dy}\right]\\ V=\pi {\left[\frac{1}{4}{y}^4\right]}_0^2\\ V=\pi {\left[\frac{16}{4}-0\right]}_0^2\\ V=4\pi \end{array}$