Geometry tells you how to figure the volumes of simple solids. Integration enables you to calculate the volumes of an endless variety of much more complicated shapes. The stack-of-pancakes technique works when the cross-sections of the 3-D shape in question are all circles.

Here’s how it works. Find the volume of the solid — between x = 2 and x = 3 — generated by rotating the curve


about the x-axis.

  1. Determine the area of any old cross section or representative pancake.

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

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


Just add some maple syrup and dig in!