Chapter 8.3, Problem 35E

To determine

To find : the volume of the solid.

Answer to Problem 35E

The volume of the solid is $8\pi$ .

Explanation of Solution

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

Calculation :

  $\begin{array}{l}y=x\\ y=-\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ x=2\end{array}$

Region specified rotate this around the y -axis.

  $V=2\pi {\displaystyle \underset{0}{\overset{2}{\int }}x\left(y_1-y_2\right)}dx$

Shell formula: $V={\displaystyle \underset{0}{\overset{2}{\int }}2\pi rh}dx$

Radius: $x$

Height: $y_1-y_2$

The volume of the solid:

  $\begin{array}{l}V=2\pi {\displaystyle \underset{0}{\overset{2}{\int }}x\left(x+\frac{x}{2}\right)}dx\\ V=2\pi {\displaystyle \underset{0}{\overset{2}{\int }}\left(x^2+\frac{x^2}{2}\right)}dx\\ V=2\pi {\left[\frac{x^3}{3}+\frac{x^3}{6}\right]}_0^2\\ V=2\pi \left(\frac{{\left(2\right)}^3}{3}+\frac{{\left(2\right)}^3}{6}\right)\\ V=8\pi \end{array}$