Question

Find the volume between z=x^2+y^2 and z=8-x^2-y^2 in two ways: by switching to cylindrical coordinates and by switching to spherical coordinates.

Answer 1

Okay. For cylindrical go ahead and convert it over giving:
\[z=r^2; z=8-r^2\]
So your z ranges from:
\[r^2 \le z \le 8-r^2\]
Then let z=0 giving r ranging from:
\[0 \le r \le 2\sqrt{2}\]
Then theta would be from:
\[0 \le \theta \le 2\pi\]
I believe.

Answer 2

Thanks so much! How do I go about switching to spherical coordinates?

Answer 3

The spherical is a little more difficult and I'm only going to be on here for a few more seconds. You need to find your bounds of rho convert your z=eq to find rho. Then you need to find phi then theta is:
\[0 \le \theta \le 2\pi\]