Find the Volume of a Solid Using the Disk Method
When the crosssections of a solid are all circles, you can divide the shape into disks to find its volume. Here’s how it works. Say you need to find the volume of a solid — between x = 2 and x = 3 — generated by rotating the curve y = e^{x} about the xaxis (shown here).

Determine the area of any old cross section.
Each cross section is a circle with radius e^{x}. So, its area is given by the formula for the area of a circle,
Plugging e^{x} into r gives you

Tack on dx to get the volume of an infinitely thin representative disk.

Add up the volumes of the disks from 2 to 3 by integrating.
A representative disk is located at no particular place. Note that Step 1 refers to “any old” cross section. It’s called that because when you consider a representative disk like the one shown in the figure, you should focus on a disk that’s in no place in particular. The one shown in the figure is located at an unknown position on the xaxis, and its radius goes from the xaxis up to the curve y = e^{x}. Thus, its radius is the unknown length of e^{x}. If, instead, you use some special disk like the leftmost disk at x = 2, you’re more likely to make the mistake of thinking that a representative disk has some known radius like e^{2}.