The problem is to find the volume of the solid generated by revolving the region bounded by the curve $x^2 - y^2 = 9$ and the lines $y=\pm4$ about the $y$-axis, using both the disk/washer and cylindrical shell methods.
(Image of problem source: https://i.stack.imgur.com/F1lqn.png)
I've found the answer using the washer/disc method but I'm at a loss as to how to find it using the cylindrical shell method. Any help would be appreciated!
The disk method works very easily in this case because the intersection of the line $y=3$ and your region generates a single disk, and the disk has the same formula for $y=1$ or $y=-3$ as for $y=3$.
Constructing the cylindrical shells is just a little more complicated, but not too complicated if you break the problem down into simpler pieces.
Consider the line $x=1$, which generates an infinite cylinder when you rotate it around the $y$ axis. What is the intersection of $x=1$ with the region you described? Does it also generate a cylinder?
What happens if you look at the line $x=4$ instead?
Not every integration problem is solved by integrating a single, simple formula over a single interval. Sometimes you have to do one integral over one interval and then a different integral over another integral. A cylindrical shell method could give you one formula when $a < x < b$ and a different formula when $c < x < d$. In some cases it may even give you two or more cylindrical shells at the same value of $x$ (but different ranges of $y$ coordinates, of course).
Disk methods can be complicated that way too even though this problem wasn't complicated to solve with disks. Rotating some other region around the $y$ axis you could have had washers with concentric disks (or other washers) in their "holes" for some values of $y$ and just disks (or washers) for other values of $y$. It often helps to draw a graph of the region that you rotated and see how various vertical or horizontal lines intersect it.