Question

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. xy = 9, x = 0, y = 9, y = 11

Answer 1

When integrating using shells, the first step is to plot the graph. I personally plotted it with the x and y axis switched, because it aids me in picturing the graph. The integral ranges from 9 to 11, since those are the limits from the two lines y = 9 and y = 11. The reason that these are the limits is because it is rotating around the x, not the y.
$\int\limits^{11}_{9}$
Now that you have the limits of the integral, you have to find what goes inside it. 
Because you are integrating using shells, you need to remember to include the $2\pi y$ Again, there is a y here instead of an x, because you are rotating around the x axis. Then you just need to input the function f(y). If you look at the graph that you (hopefully) plotted, you can see that this function ranges between the y axis and the curve $f(y) = \frac{9}{y}$. Put together the pieces, and you have the integral
$\int\limits^{11}_{9} {2 \pi y*f(y)} \, dy$
After substituting in $f(y) = \frac{9}{y}$, you get 
$\int\limits^{11}_{9} {2 \pi y*\frac{9}{y}} \, dy$
Simplified, this is $18\pi\int\limits^{11}_{9} \, dy$
Integrating, we get $18\pi * 2$

Therefore, the solution is $36 \pi$

Note: I didn't spend very much time reviewing these integrals, so I may be incorrect.