If I am given an area characterised by $0 \leq z \leq 1-r^{2}$ from this how can I deduce the radius r and the angle with the x-axis, $\theta$ that will span the are and I can ten integrate over i.e. find the limits of integration.
The original area of integration was in cartesian coordinates and I have converted to cylindrical.
$z$ varies from $0$ to $1-r^2$. In cartesian, $1-r^2=1-x^2-y^2=1-(x^2+y^2)$. From this form, I hope it's clear that it's a downward opening paraboloid, with a vertex at the point $(0,0,1)$. So the limits of integration is where that paraboloid intersects the plane $z=0$. Does that help?