How do you know when to use the shell method or the disk method?

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 #x=0#, #y=x^2#, and #y=-x+2# around the #y#-axis, I would use shells because there would only be one integral to evaluate. (Disks would require two: one from #y=0# to #y=1# and another from #y=1# to #y=2#.)

Taking #y=0#, #y=x^2#, and #y=-x+2# around the #x#-axis, I would use shells to avoid doing two integrals even though it would require me to rewrite the curves as functions of #y#.

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 #y=x^4-8x^3+x+22# we do not want to try to rewrite this as a function of #y#. So we'll use the set-up that keeps things in terms of #x#.