Find the Volume of a Solid Using the Disk Method

By Mark Ryan

When the cross-sections 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 = ex about the x-axis (shown here).

A sideways stack of disks.
A sideways stack of disks.
  1. Determine the area of any old cross section.

    Each cross section is a circle with radius ex. So, its area is given by the formula for the area of a circle,

    image1.png

    Plugging ex into r gives you

    image2.png

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

    image3.png

  3. Add up the volumes of the disks from 2 to 3 by integrating.

    image4.png

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 x-axis, and its radius goes from the x-axis up to the curve y = ex. Thus, its radius is the unknown length of ex. If, instead, you use some special disk like the left-most disk at x = 2, you’re more likely to make the mistake of thinking that a representative disk has some known radius like e2.