**Volume by rings**, also known as **volume by disks** or **volume by washers** (if the area between two functions is being rotated around an axis), is a method of finding the volume of a solid of revolution. This method involves splitting the shape into indefinitely small circular rings. The formula for the volume of any solid of rotation is or , depending on which axis the rotation is around. In the case of volume by rings, the formula is

or

assuming the rotation is around the x-axis. If the rotation is of an area between two functions, the formula is

## Examples

To find the volume of the resulting solid when is rotated around the -axis on the interval , substitute into the formula.

This method can also be used to find the formula for the volume of shapes. Take for example the formula for the volume of a sphere, . A sphere is a graph of rotated around an axis (here we will assume it is the -axis). Begin by isolated . It is important to keep in mind that is not a variable and must not be treated as one.