# Volume of a Sphere

Let's say that we want to find the volume of a sphere of radius  $r$ using volumes of revolution.

We know that the equation of a circle of radius  $r$ centered at the origin is

$x^{2}+y^{2}=r^{2}.$ The upper half semicircle is given by  $y={\sqrt {r^{2}-x^{2}}}.$ Now, we want to rotate the upper half semicircle around the  $x$ -axis. This will give us a sphere of radius  $r.$ We use the washer/disk method to find the volume of the sphere. The volume of the sphere is

${\begin{array}{rcl}\displaystyle {V}&=&\displaystyle {\int _{-r}^{r}\pi ({\sqrt {r^{2}-x^{2}}})^{2}~dx}\\&&\\&=&\displaystyle {\int _{-r}^{r}\pi (r^{2}-x^{2})~dx}\\&&\\&=&\displaystyle {\pi {\bigg (}r^{2}x-{\frac {x^{3}}{3}}{\bigg )}{\bigg |}_{-r}^{r}}\\&&\\&=&\displaystyle {\pi {\bigg (}r^{3}-{\frac {r^{3}}{3}}{\bigg )}-\pi {\bigg (}-r^{3}+{\frac {r^{3}}{3}}{\bigg )}}\\&&\\&=&\displaystyle {{\frac {4}{3}}\pi r^{3}.}\end{array}}$ Hence, the volume of a sphere of radius  $r$ is

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