As volume is the antiderivative of area, the integral can be used to find the volume, *V*, of an *integrated* "family" of disks. The necessary equation, for calculating such a volume, is slightly different depending on which axis is serving as the axis of revolution. These equations note that the area of a disk (one which has no height, and no volume) equals: pi (π) multiplied by the disk's squared radius (*r*^{2}). The volume of a representative disk equals: π*r*^{2}which is in turn multiplied by the disk's length (or height), *dy*, that being some number approaching zero.

- Horizontal Axis of Revolution
*V***=**∫ [*π*(*R*)]*x*^{2}*dy*

- Vertical Axis of Revolution
*V***=**∫ [*π*(*R*)]*y*^{2}*dy*

Consider the region bounded by *R*(*x*) = √*x* and *r*(*x*) = *x*^{2}. One can calculate the volume of a sphere of revolution, which has a radius of √*x*, as shown above -- π ∫ (√*x*)^{2} *dx* = π / 2. One should then calculate the volume of the "inner" sphere of revolution, that having a radius of *x*^{2} -- π ∫ (*x*^{2})^{2} *dx* = π / 5. By subtracting the inner area, π / 2 - π / 5, one obtains the volume of the bounded area: 3π / 10.

see also: shell integration