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How to Find the Volume of a Circular Shape with the Stack-of-Pancakes Method of Integration

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

image0.png

about the x-axis.

image1.jpg
  1. Determine the area of any old cross section or representative pancake.

    image2.png
  2. Tack on a dx to get the volume of an infinitely thin representative pancake.

    image3.png
  3. Add up the volumes of the pancakes from 2 to 3 by integrating.

    image4.png

Just add some maple syrup and dig in!

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