Question

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

Answer 1

Hint: As we know that these are the two methods that are used in the application of integrations. We know that a solid of revolution is defined as a solid created by revolving a region about an axis, one that is ideally external to the said region. We can find the volume of such a solid can be determined by the Shell method or the Disk method. Disk method can also be called a washer method.

Complete step by step solution:
As we know that the disk method is typically easier when evaluating revolutions around the $x - axis$, whereas we can say that the shell method is easier for revolutions around the $y - axis$, especially for the final solid when it has a hole in it.
We can say that the disk method can be written as :
$V = \pi \int\limits_a^b {r{{(x)}^2}.{dx}} $.
While the shell or washer method can be written as :
$V = 2\pi \int\limits_a^b {xf{{(x)}.{dx}}} $.
Hence this is the main difference between both the methods.

Note: We should note that the disk method is about stacking disks of varying radii and shape which is defined by the revolution of $r(x)$ along the x- axis at each $x$, while the shell method is about vertically layering rings which is defined by $2\pi x$ of varying thickness and shape $f(x)$.