Prove $\int_{-1}^{1}\int_{-1}^{1}\left(1-x^2-y^2\right)^{\left(n/2\right)-1} {\rm d}x\,{\rm d}y = 2\pi/n$ As the title says, I want to show that
\begin{align}
&\int_{-1}^{1}\int_{-1}^{1}\left(1 - x^{2} - y^{2}\right)^{\left(n/2\right) - 1}\,
{\rm d}x\,{\rm d}y = 2\pi/n
\\&\left(~\mbox{this came up in finding a recursion for the volume of}\
n\mbox{-dimensional spheres}~\right)
\end{align}
I feel like a substition might work here. Does anyone have a hint for me ?.

Yes. When you are integrating with respect to $x,$ $y$ is a constant, so you just need to integrate $\sqrt{a^2 - x^2}.$ Can you?
