…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
[1] 0.3273202
attr(,"error")
[1] 2e-16
attr(,"status")
[1] "normal completion"