The integral $M_2$ was studied in a US military report, to calculate the probabability of a missile hitting an elliptical target. See equation 31, where the integral $M_2=p(a,b)$ is given as a sum over Bessel functions $I_n$. (There does not seem to be a closed form solution.)

The integral $M_2$ was studied in a 1961 US military report, to calculate the probabability of a missile hitting an elliptical target. See equation 31, where the integral $M_2=p(a,b)$ is given as a sum over Bessel functions $I_n$. (There does not seem to be a closed form solution.)

The answer can be seen as a two-dimensional analogue of the error-function, defined as an integral over $I_0$,

$$E(R,r)=e^{-r^2/2}\int_0^R e^{-t^2/2}I_0(rt)tdt$$

Then the desired integral over the ellipse is

$$M_2=\frac{1}{2\pi}\int\int_{x^2/a^2+y^2/b^2\leq 1} e^{-(x^2+y^2)/2}dxdy$$ $$=E[(a+b)/2,(a-b)/2]-E[(a-b)/2,(a+b)/2]$$