Derivation of Formula for Volume of the Sphere by Integration

For detailed information about sphere, see the Solid Geometry entry, The Sphere.

The formula for the volume of the sphere is given by

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

Where, r = radius of the sphere

Derivation for Volume of the Sphere
Figure for the Derivation of Formula of Sphere by IntegrationThe differential element shown in the figure is cylindrical with radius x and altitude dy. The volume of cylindrical element is...
$dV = \pi x^2 dy$

The sum of the cylindrical elements from 0 to r is a hemisphere, twice the hemisphere will give the volume of the sphere. Thus,
$\displaystyle V = 2\pi \int_0^r x^2 dy$

From the equation of the circle x2 + y2 = r2; x2 = r2 - y2.

$\displaystyle V = 2\pi \int_0^r (r^2 - y^2) dy$

$V = 2\pi \left[ r^2y - \dfrac{y^3}{3} \right]_0^r$

$V = 2\pi \left[ \left(r^3 - \dfrac{r^3}{3}\right) - \left(0 - \dfrac{0^3}{3}\right) \right]$

$V = 2\pi \left[ \dfrac{2r^3}{3} \right]$

$V = \dfrac{4 \pi r^3}{3}$       okay!