# How to find the volume of a sphere using integration?

##### 1 Answer
Sep 18, 2014

Since a sphere with radius $r$ can be obtained by rotating the region bounded by the semicircle $y = \sqrt{{r}^{2} - {x}^{2}}$ and the x-axis about the x-axis, the volume $V$ of the solid can be found by Disk Method.

$V = \pi {\int}_{- r}^{r} {\left(\sqrt{{r}^{2} - {x}^{2}}\right)}^{2} \mathrm{dx}$

by the symmetry about the y-axis,

$= 2 \pi {\int}_{0}^{r} \left({r}^{2} - {x}^{2}\right) \mathrm{dx}$

$= 2 \pi {\left[{r}^{2} x - {x}^{3} / 3\right]}_{0}^{r}$

$= 2 \pi \left({r}^{3} - {r}^{3} / 3\right)$

$= \frac{4}{3} \pi {r}^{3}$