If I have a 2D region defined as follows:
\begin{align} &(1) \qquad u^2-v^2 \leq a \\ &(2) \qquad u^2-v^2 \geq -a \\ &(3) \qquad u \geq 0 \\ &(4) \qquad v \geq 0 \end{align}
and I want to take a double integral over this region, are my bounds correct?
$$\int_0^{\sqrt{a}} dv \int_0^{\sqrt{v^2+a}} du ~ F(u,v)+\int_{\sqrt{a}}^\infty dv \int_{\sqrt{v^2-a}}^{\sqrt{v^2+a}} du ~ F(u,v)$$
If not, what's my mistake?
I need this to solve a PDE in parabolic coordinates.