Volume of Sphere/Proof by Method of Disks

Theorem
The volume $V$ of a sphere of radius $r$ is given by:
 * $V = \dfrac {4 \pi r^3} 3$

Construction
Describe a circle on the $xy$-plane.

Let its center be the origin.

By Equation of a Circle, this circle is the locus of:
 * $x^2 + y^2 = r^2$

where $r$ is a constant radius.

Solving for $y$:
 * $y = \pm \sqrt {r^2 - x^2}$

Considering only the upper half of the circle:
 * $y = \sqrt {r^2 - x^2}$


 * Semicircle.png

This plane region is a semicircle whose radius is $r$ and whose extremes are at $x = -r$ and $x = r$.

By Euclid's definition of a sphere, the solid of revolution of this plane region about the $x$-axis is a sphere whose radius is $r$.

Proof
Note that this proof uses the Method of Disks and thus is dependent on Volume of a Cylinder.

From the Method of Disks, the volume of this sphere can be found by the definite integral:


 * $\displaystyle V = \pi \int_{-r}^{r} y^2 \ \mathrm d x$

where $y$ is the function of $x$ describing the curve which is to be rotated about the $x$-axis in order to create the required solid of revolution.

By construction, $y = \sqrt {r^2 - x^2}$.

The volume, then, is given by: