Chapter 8, Problem 45RE

To determine

To find : the volume of the solid.

Answer to Problem 45RE

The volume of the solid is $3\pi$ .

Explanation of Solution

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

Calculation :

From the cylindrical shell method:

  $V={\displaystyle \underset{a}{\overset{b}{\int }}2\pi xf\left(x\right)dx}$

With $f\left(x\right)$ the function that bounds the region is bounded.

  $\begin{array}{l}V={\displaystyle \underset{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\overset{2}{\int }}2\pi x\left(\frac{1}{x}\right)}dx\\ V=3\pi \end{array}$