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.

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 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 . Put together the pieces, and you have the integral

After substituting in , you get

Simplified, this is

Integrating, we get

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 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 . Put together the pieces, and you have the integral

After substituting in , you get

Simplified, this is

Integrating, we get

Therefore, the solution is

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