Definite integral of two dimensional normal distribution

Question

I want to calculate the probability mass in a rectangular area for a two dimensional normal distribution $G(x;\mu,\Sigma)$ . Is it possible to do without numerical integration?

Answer 1

You can write a "closed form" expression for the bivariate normal integral using the Owen's T function (see references therein). Even in the univariate case, the normal integral has no closed form solution but its value can be written in terms of other standard functions.