6.4 Volume by Shells

There are many problems where the disk & washer method is perfectly effective. But in some instances you’ll need to know about the shell method for computing the volume of a solid of revolution.

This video really demonstrates how shells work:

The TL;DR is that it is easier to use the washer method for finding the volume of a solid made by rotating about the x-axis, but the shell method is easier to use to find the volume of a solid made by rotating about the y-axis. This video should further show you the difference between the washer method vs. the shell method:


Volume by the Shell Method Formula

Let f and g be continuous functions with f(x) ≥ g(x) on [a, b]. If R is the region bounded by the curves y = f(x) and y = g(x) between the lines x = a and x = b, the volume of the solid generated when R is revolved about the y-axis is:

You can also conceptually understand the shell method formula as ∫2π(Shell Radius)(Shell Height)dx

Note that the shell method is meant for the y-axis. If you revolve the shell around the x-axis, then you basically slip everything around.


What's Your Opinion?