# How do you find the volume of the region enclosed by the curves y=2x, y=x, and y=4 is revolved about the y-axis?

Oct 21, 2015

Volume$= 16 \pi$

#### Explanation:

The region described is the difference between two inverted cones as can be seen from the diagram below:

The volume of the bounded region is the difference between the volume of the external cone and the internal cone.

The general formula for the volume of a cone is
$\textcolor{w h i t e}{\text{XXX}} V = \frac{\pi {r}^{2} h}{3}$

So the required region has a volume of
$\textcolor{w h i t e}{\text{XXX}} \frac{\pi \cdot {4}^{2} \cdot 4}{3} - \frac{\pi \cdot {2}^{2} \cdot 4}{3}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{4 \pi}{3} \cdot \left(16 - 4\right)$

$\textcolor{w h i t e}{\text{XXX}} = 16 \pi$