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!
 

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