Volume of a Sphere

Let's say that we want to find 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}   using volumes of revolution.

We know that the equation of a circle 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}   centered at the origin 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 x^2+y^2=r^2.}

The upper half semicircle is given by  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 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

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.}