I am working on a problem that requires me to find the volume of the solid bounded by the sphere $x^2 + y^2 + z^2 = 2$ and the paraboloid $x^2 + y^2 = z$. I know that to do this, I must use triple integration.
Thus far, I know that we need to reevaluate the given equations and put them into spherical coordinates in order to achieve our bounds. When we've gotten our bounds, it's just simple triple integration to computer the volume.
So, for our sphere, I said that $x^2 + y^2 + z^2 = 2$ is equivalent to $\rho = sqrt(2)$.
Then, for our sphere, I get a little messed up. I'm trying to find $\phi$ boundaries here, but I get the equation $\rho^2sin^2\phi = \rho cos\phi$ Then, the $\rho$ can cancel out, making it $\rho sin^2\phi = cos\phi$, but my issue is that the $\rho$ doesn't cancel out completely so I can't get the bounds for $\phi$.
I'm very new to doing this type of thing, so any help would be greatly appreciated. Anything I'm missing or didn't know about? Thank you very much!