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: $y={\sqrt {r^{2}-x^{2}}}.$

(insert picture of semicircle)

Now, we want to rotate the upper half semicircle around the $x$-axis. This will give us a sphere of radius $r.$

(insert pictures)

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