Finding the volume between a cylinder and a hyperboloid

Question

I'm having trouble coming up with the limits needed to do the integration. The equation of the cylinder is $y^2 + z^2 = 1$, and the equation for the hyperboloid is $z = \sqrt{a^2 + x^2 + y^2}$ with $ 0<a<1$. So far I have z is between a and 1, $\theta$ is between $0$ and $2\pi$, but I can't work out what $r$ is meant to be between.

Answer 1

I think it helps to find the intersection, so we know that $z>0$: $$z^2=a^2+x^2+y^2$$ $$1-y^2=a^2+x^2+y^2$$ $$1-a^2=2x^2+y^2$$ and the solutions to this equation form an elipse of the form: $$(\sqrt{2}x)^2+y^2=(1-a^2)$$ so this gives us the upper limits for $x$ and $y$ and $z$ will vary from $a$ to $1$