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.
[tex] \int\limits^{11}_{9}[/tex]
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 [tex]2\pi y[/tex] 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 [tex]f(y) = \frac{9}{y}[/tex]. Put together the pieces, and you have the integral
[tex] \int\limits^{11}_{9} {2 \pi y*f(y)} \, dy [/tex]
After substituting in [tex]f(y) = \frac{9}{y}[/tex], you get
[tex] \int\limits^{11}_{9} {2 \pi y*\frac{9}{y}} \, dy [/tex]
Simplified, this is [tex] 18\pi\int\limits^{11}_{9} \, dy [/tex]
Integrating, we get [tex] 18\pi * 2[/tex]
Therefore, the solution is [tex] 36 \pi [/tex]
Note: I didn't spend very much time reviewing these integrals, so I may be incorrect.
