# What is the volume of the solid produced by revolving f(x)=7-x, x in [0,2] around the x-axis?

Feb 27, 2017

$V = \frac{218 \pi}{3}$
$\setminus \setminus \setminus \approx 228.289$

#### Explanation:

graph{7-x [-10.29, 9.71, -1.88, 8.12]}

$y = 7 - x$ is a straight line; The volume of revolution is that of a large cone of radius $7$ height $7$ minus that of smaller cone radius $5$ height $5$; Using ${V}_{\text{cone}} = \frac{1}{3} \pi {r}^{2} h$, Then:

$V = \frac{1}{3} \pi {\left(7\right)}^{2} \left(7\right) - \frac{1}{3} \pi {\left(5\right)}^{2} \left(5\right)$
$\setminus \setminus \setminus = \frac{1}{3} \pi \left({7}^{3} - {5}^{3}\right)$
$\setminus \setminus \setminus = \frac{1}{3} \pi \left(343 - 125\right)$

$\setminus \setminus \setminus = \frac{218 \pi}{3}$
$\setminus \setminus \setminus \approx 228.289$