Method of disks or the method of rings

One of the simple methods for getting the cross-sectional area is to cut the object perpendicular to the axis of rotation.  Carrying out this the cross section will be either a solid disk if the object is solid (as our above example is) or a ring if we've hollowed out portion of the solid (we will illustrates this eventually).

In the case that we obtain a solid disk the area is,

A = ∏ ( radius )²

where the radius will based upon the function and the axis of rotation.

In the case that we get a ring the area is following,

where again both of the radii will based on the functions given & the axis of rotation.  Note that in the case of solid disk we can think of the inner radius as zero & we'll arrive at the correct formula for solid disk and therefore this is a much more general formula to utilize.

Also, in both of the cases, whether the area is a function of x or a function of y will based upon the axis of rotation as we will illustrates.

This method is frequently called the method of disks or the method of rings.