Shells are used when we slice parallel to the axis of rotation (remember paraSHELL)

General Notes

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The most important steps of finding volume using cylindrical shells is:

  1. Determine the axis of revolution. Rotation along y axis means vertical slices, and the variable of integration is x
  2. Define shell components. Find the radius which is the distance from the axis of rotation to the slice. Height is length of the slice or top-bottom or right-left
  3. Setup the integral using the known bounds, radius and height

For the shell method, the region can either touch or not touch the axis. The only dependence on the shell method is the fact the slices are parallel to the axis of rotation

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(From Mathematics LibreTexts)

Practice Problems

https://tutorial.math.lamar.edu/Problems/CalcI/VolumeWithCylinder.aspx

https://tutorial.math.lamar.edu/Problems/CalcI/VolumeWithCylinder.aspx