# Question #8f07d

Mar 27, 2017

$V = 36 \pi$

#### Explanation:

Consider the line:

$y = \frac{x}{4}$

For $x = 12$ we have $y = 3$, so we can see that rotating this line around the $x$-axis in the interval $x \in \left(0 , 12\right)$ generates exactly the solid requested.

The element of volume generated by the cylindrical shell between $x$ and $x + \mathrm{dx}$ is given by:

$\mathrm{dV} = \pi {y}^{2} \left(x\right) \mathrm{dx} = \pi {x}^{2} / 16 \mathrm{dx}$

so, integrating over the interval:

$V = \frac{\pi}{16} {\int}_{0}^{12} {x}^{2} \mathrm{dx} = \frac{\pi}{16} {\left[{x}^{3} / 3\right]}_{0}^{12} = 36 \pi$