Q:  When should I use a disk / washer method versus a shell / cylinder method for integration?

Answer:

First, the visual difference: 

The disk / washer method is used when you can think of your shape as “stacked pancakes” (the washer method is just removing any center “holes” from these pancakes).

The shell / cylinder method is used when you can think of your shape as “stacked Russian Dolls” (you know, those dolls that stack inside of each other, and you keep opening them up to find a small doll inside, etc..)

A mathematical difference:

Example 1:  Rotate around the x-axis to create a volume.

If you rotate around the x-axis and use the disk method:  You will be stacking your pancakes horizontally (with respect to x).  Therefore, your limits and functions of integration will be in terms of “x” (it will be a “dx” problem).  The radius of your disk will need to be in terms of x.

If you rotate around the x-axis and use the shell method:  You will be stacking your cylinders vertically.  Your limits and functions of integration will all be in terms of “y” (it will be a “dy” problem).  The height of your cylinder will need to be in terms of y.  The radius of your cylinder will most likely just be “y” itself – though not guaranteed.

Example 2:  Say you are rotating an area around the y-axis to create a volume.

If you rotate around the y-axis and use the disk method:  You will be stacking your pancakes vertically (with respect to y).  Therefore, your limits and functions of integration will be in terms of “y” (it will be a “dy” problem).  The radius of your disk will need to be in terms of y.

If you rotate around the y-axis and use the shell method:  You will be stacking your cylinders horizontally.  Your limits and functions of integration will all be in terms of “x” (it will be a “dx” problem).  The height of your cylinder will need to be in terms of x.  The radius of your cylinder will most likely just be “x” itself – though not guaranteed.

Which method to pick?

Take into account the visual of your shape.  Does it look like pancakes stacked on top of each other or cylinders nested inside of each other?

Take into account the math.  Is it easier to put the functions in terms of y or in terms of x?  That will guide you on which to pick (depending on your axis of rotation).

This was a start to understanding.  Next read: When to use Disk Method versus Shell Method, Part 2.

Also check out this example…  I do one problem using the disk method.  Then, I do the same problem using the shell method:  Integration Example: Disk Method vs. Shell Method