Method of cylinders or method of shells

The formula for the area in all of the cases will be,

                                                       A = 2 ∏ ( radius ) (height )

There are some important differences among this method & the method of rings/disks that we have to note before moving on.  Firstly, rotation around a vertical axis will give an area which is a function of x and rotation around a horizontal axis will give an area which is a function of y. It is exactly opposite of the method of rings/disks.

Secondly, we don't take the total range of x or y for the limits of integration.  Rather than we take a range of x or y which will cover one side of the solid. If we expand out the radius to cover one side automatically we will expand in the other direction as well to cover the other side.