asymptotics - Approximating a multiple sum with an integral

Hi,

I want to approximate a multiple sum of the form
$$sum_{x_1+x_2+cdots+x_m leq n}e^{g(x_1,x_2,ldots,x_m)},$$
where each $x_i$ is an integer between $0$ and $n$,
by an integral
$$int_{x_1+x_2+cdots+x_m leq n}e^{g(x_1,x_2,ldots,x_m)}dx_1dx_2cdots dx_m,.$$
I know that the Euler-Maclaurin formula can be used to derive the error term when $m=1$ but often see sums of this form with $m > 1$ approximated by integrals, though with little justification. I do not have much of a background in mathematical analysis so am not sure where to look for a reference for this.

Any help will be much appreciated.