How do you know when to use the shell method or the disk method?
You'll get the same answer either way.
Sometimes one leads to an integral that a particular person finds easier to evaluate, but what is easier varies between people.
I have a preference for doing a single integral.
So if I have to find the volume of the solid generated by revolving the region bounded by
Again, that is my preference. As a student, perhaps because we learned disks/washers first, I preferred them.
In the end it doesn't matter. If you set it up one way and don't care for the looks of the integral, try setting it up the other way.
It is possible that you need to use a function whose inverse you can't find. For instance if one boundary of the region is