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How to Find Volume by the Shell Method

One element of calculus involves figuring out the volume of the shells created when two-dimensional objects are turned around an axis of revolution. Imagine a rectangle of length "L" and width "W," and then an axis that is "A" units away from the center of that rectangle. Spinning that rectangle around that axis would create a hollow cylinder, and there is a specific way to find the formula of that cylindrical shell.

Instructions

- 1
  Add "A" to half the width ("W") of the rectangle. Square that sum.
- 2
  Multiply the answer from Step 1 by the length ("L") of the rectangle, and multiply that answer by pi. Now, you have the volume of the whole cylinder. Since you're looking at a rotated shell, though, you have to subtract the inner cylinder's volume.
- 3
  Subtract half of the rectangle's width ("W") from the distance "A." Square that difference.
- 4
  Multiply the answer to Step 3 by the length of the rectangle ("L"), and then multiply that answer by pi.
- 5
  Subtract the answer to Step 4 from the answer to Step 2.

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