Given a bivariate gaussian distribution (see below) with unequal $\sigma_x, \sigma_y$, how do I compute the integral within a constant height contour?
If I'm not mistaken, the following bivariate gaussian should be a proper normal distribution so the integral is 1 when the contour ellipse is infinitely large:
$\displaystyle f(x,y) = \frac{1}{2\pi\sigma_x\sigma_y}{\exp({-\frac{1}{2}(\frac{x^2}{\sigma_x^2}+\frac{y^2}{\sigma_y^2}}}))$