Double Integrals · Challenge problem

Using polar coordinates, the integral $\displaystyle\iint_D e^{-(x^2+y^2)}dA$ over the WHOLE plane equals $\pi$. Given that, the famous one-dimensional integral $\int_{-\infty}^{\infty} e^{-x^2}dx$ equals $\sqrt{k\pi}$ with $k = 1$... instead: compute $\displaystyle\iint_D 1\,dA$ where $D$ is the quarter disk $x^2+y^2\le 4$, $x,y \ge 0$. The answer is $k\pi$; find $k$.