…in R of course!
There is a handy function to do those calculations. Normally (ahh!) you might resolve to a symbolic calculation package (Maple,Mathematica etc.) but that is not the situation any more. The calculations are done with the mnormt package. Relevant functions exist in other packages as well (R : Distributions)
x <- seq(-2,4,length=21) y <- 2*x+10 z <- x+cos(y) mu <- c(1,12,2) Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3) p2 <- sadmvn(lower=rep(-Inf,2), upper=c(2, 11), mu[1:2], Sigma[1:2,1:2]) > p2  0.3273202 attr(,"error")  2e-16 attr(,"status")  "normal completion"