I am looking for a really fast way to integrate numerically the 2-dimensional gaussian density with identity covariance matrix $\Sigma=\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)$ and some fixed and known mean value $\mu\in\mathbb{R}^2$ over the region of the plane described in polar coordinates by $$ \{(r,\phi)\colon r\in[0,+\infty), \phi\in[\phi_1,\phi_2]\}, $$ (where $\phi_1,\phi_2$ are known), which is the "angle" between two half-lines starting from the origin. I need to evaluate these integrals for different values of $\phi_1,\phi_2$ (different pairs of half-lines) $n(n-1)/2$ times.
I failed to get the closed-form expression for this integral, so i am now using MATLAB function "quad2d". For $n=100$ it takes about 1 minute. And i need to compute it for $n\approx 10^4$.
Any ideas?
Thanks alot!