If you have , then the volume can be calculated as
where is
It comes from the fact that
My Prof. probably didn't think this one through too much. I found a similar question to this one here, yet from it I discerned an issue with the question (which is listed below, word-for-word as usual): it doesn't state which surface the region is bounded above, and which surface the region is bounded below. Anyhow, here is the question.
Find the volume of the region between the surfaces and .
(I know, the wording is horrible; so open to loopholes)
All I really know is that I have to get to here somehow:
?
So far, here is what I have, borrowing bits from the link above.
For the point of intersection between the two surfaces, setting (I think), I got the following:
From here, however, I'm not sure what to do. It's very little, I know, but I'm just not getting this material drilled in right because of a lack of legible examples.
Okay, I've figured this one out thanks to another source. Here's my answer (I might've skipped a step or two, or gotten a step mixed up):
First, we determine the point of intersection between the two surfaces:
, which is the region .
The surfaces intersect on the cylinder .
As such, the top surface is , and the bottom surface is .
Thus, the integrand is as follows:
Converting to polar coordinates, the integrand becomes the following:
Evaluating at , we integrate and obtain the following:
Therefore, the volume of the region is .