Question

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

Answer 1

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 = piint_a^b (r(x))^2dx`

The shell method is:

`V = 2piint_a^b xf(x)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(x)` along the x-axis at each `x`), the shell method is about vertically layering rings (defined by `2pix`, where `x` is the radius of the ring) of varying thickness and shape `f(x)`.