Proof of volume of sphere

The volume of a sphere is the integral of an infinite number of infinitesimally small circular slabs of thickness dx. The calculation for the volume of a sphere with center 0 and radius r is as follows.

The surface area of the circular slab is .

The radius of the circular slabs, defined such that the x-axis cuts perpendicularly through them, is;

or

where y or z can be taken to represent the radius of a slab at a particular x value.

Using y as the slab radius, the volume of the sphere can be calculated as

Now

Combining yields gives

This formula can be derived more quickly using the formula for the sphere's surface area, which is . The volume of the sphere consists of layers of infinitesimally small spherical slabs, and the sphere volume is equal to

=