# How do you find the volume of the solid y=4-x^2 revolved about the x-axis?

May 20, 2017

Use the disc method. The volume is $107.233$.

#### Explanation:

First, set $y$ equal to $0$ to find the bounds.

$0 = 4 - {x}^{2}$
${x}^{2} = 4$
$x \in \left\{- 2 , 2\right\}$

So our bounds are $- 2$ and $2$.

Next, use the disc method to find the volume ($y = r$).

${\int}_{-} {2}^{2} \pi {y}^{2} \mathrm{dx} = \pi {\int}_{-} {2}^{2} {\left(4 - {x}^{2}\right)}^{2} \mathrm{dx}$

$= \pi {\int}_{-} {2}^{2} \left(16 - 8 {x}^{2} + {x}^{4}\right) \mathrm{dx}$

$= \pi {\left[16 x - \frac{8}{3} {x}^{3} + {x}^{5} / 5\right]}_{-} {2}^{2}$

$= \pi \left(32 - \frac{64}{3} + \frac{32}{5}\right) - \pi \left(- 32 + \frac{64}{3} - \frac{32}{5}\right)$

$= \frac{512}{15} \pi$

$= 107.233$