| The key is the representative cross-section:
If your cross-section lies parallel to the axis of revolution, shells will work. If your cross-section lies perpendicular to the axis of revolution, discs/washers will work.
The direction of your cross-section will frequently be based on what's easier to work with. For instance, in cases where y is a function of x, you tend to use discs/washers around a horizontal line, but shells around a vertical line.
The way you get around using shells when you want to revolve around a vertical line is to solve that function for an x = equation as a relation of y.
In some cases, transferring from one form to the other is difficult, and so the AP folks have basically guaranteed that you can always use a discs/washers form, while the shells form might not always work.
That being said, if you have the time and feel comfortable with the normal stuff, shells might be easier than discs/washers on a particular problem, and some students prefer volumes by shells to discs/washers, even if the calculations aren't necessarily "easier". |