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

        Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=\frac{4}{3}\pi r^3.}