I was given the following question to solve:
"For a region bounded by the curve $x^2-y^2=9$ and the lines $y=4, y=-4$, find the volume of the solid generated by revolving the region about the x-axis.
I have found a solution through the Washer method. To verify this solution, I tried solving the problem through the cylindrical shell method as well. I could not get a solution through this method. I thus wonder whether the original solution I obtained was correct, and whether I am misunderstanding how to use the Cylindrical Shell method in this case.
This is my first time dealing with an equation where x (and also y) can be expressed in terms of two functions, so I have decided to find the volume generated by one function and multiply it by two. To make things more clear, I have included my attempted solutions below. I appreciate your guidance.
(note that the equation of interest is $x^2-y^2=9$, not $x^2+y^2=9$, sorry about that)
Best Answer
You did almost right. See below and try to understand what went wrong in your calculations.
Washer method: $$V=\pi\int_{-5}^{-3} (4^2-(x^2-9)) dx+\pi\int_{-3}^3 4^2 dx+\pi\int_{3}^{5} (4^2-(x^2-9)) dx=\frac{392\pi}{3}.$$
Cylindrical shell method: $$V=2\pi\int_{-4}^4|y|\sqrt{9+y^2}dy=\frac{392\pi}{3}.$$