I am having some trouble with this problem, I feel like I am just confusing myself and I could really use some direction.
Problem
For positive $a$ and $h$, let $A$ designate the region of $\mathbb R^3$ enclosed by the elliptic hyperboloid $x^2+y^2-z^2=a^2$, and the two planes $z=-h/2$ and $z=h/2$. Let $B$ be the orientable surface of $A$.
Determine the volume of $A$.
For the position vector field $\vec{F}=\vec{R}=\langle x,y,z\rangle$, calculate the flux out of the lateral surface of $A$.
and 4. Calculate the flux out of the top and bottom surface e of $A$.
My thoughts:
Part 1: I feel like it might be easiest to set the integral up using cylindrical or polar coordinates, I am just not feeling like I'm getting the boundaries right.
Part 2: To calculate the flux, I need to parameterize the position vector field and then do I use Stokes' Theorem? I'm not sure how to find the flux out of the lateral surface (or the bottom or top in the other parts).
I would love it if someone could do one part for me or point me to a similar example so I can see what the process looks like. I'm not sure why I'm so confused on this but there are just so many different integrals in this vector calc chapter I feel like I'm confusing them with one another.
Any help would be very much appreciated.