# What is the difference between the shell method and disk method?

Aug 28, 2015

The disk method is typically easier when evaluating revolutions around the x-axis, whereas the shell method is easier for revolutions around the y-axis---especially for which the final solid will have a hole in it (hence shell).

The disk method is:

$V = \pi {\int}_{a}^{b} {\left(r \left(x\right)\right)}^{2} \mathrm{dx}$

The shell method is:

$V = 2 \pi {\int}_{a}^{b} x f \left(x\right) \mathrm{dx}$

Another main difference is the mentality going into each of these.

While the disk method is about stacking disks of varying radii and shape (defined by the revolution of $r \left(x\right)$ along the x-axis at each $x$), the shell method is about vertically layering rings (defined by $2 \pi x$, where $x$ is the radius of the ring) of varying thickness and shape $f \left(x\right)$.