Derivation of Formula for Volume of the Sphere by Integration

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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!