I was looking through some past papers and came across the following:
Express the volume that is contained inside both the sphere x^2+y^2+z^2=4 and the cylinder x^2+y^2=1 as a triple integral in:
1. Cartesian Co-ordinates.
2. Cylindrical Polar Co-ordinates.
Then, I am asked to evaluate (2) to find the required volume.
I am unsure how to go about answering this. I thought about writing:
Then, I would have a triple integral. I would integrate with respect to z first, between -sqrt(4-x^2-y^2) and +sqrt(4-x^2-y^2). Then, y between -sqrt(1-x^2) and +sqrt(1-x^2), finally x between -2 and 2. However, should I be integrating x^2+y^2-1 here?!?
I can see how the second part helps (r between 0 and 1, theta between 0 and two Pi), but I dont know how to go about writing it without the first part. I also dont know what z would be in this case.
Any help would be appreciated, i've been scratching my head on this one the last week or so now!