I have an integral of the type: \begin{equation} I=\int_A f(x,y)\,dx\,dy \end{equation}
where \begin{equation} A=\{(x,y) : x\leq X,y\leq Y,x+y \geq W\} \end{equation}
Assuming that $f(x,y)$ has a closed-form anti-derivative, and that $A$ is not empty, I can handle this case by sketching the region $A$ (a triangle) and defining the limits of the double-integral to evaluate $I$.
With a 3-dimension problem maybe I can use the shadow method or the cross section method, but my problem is in dealing with higher dimensions, where I cannot sketch the region $A$. Being unable to define the limits of integration, the only way I know to evaluate $I$ is to use numerical integration methods (i.e. MonteCarlo), using an indicatory function $\mathbf{1}_A (x,y)$ so I can consider only values of $(x,y)$ actually in $A$.
So, I was wondering how to find the limits of integration given a set $A$ in the form: \begin{equation} A=\{(x_1,\dots,x_n) : x_i\leq X_i, \sum_{i=1}^n x_i \geq W\} \end{equation}
Since the equations defining $A$ are all linear inequalities, is there a general way to move from the set-notation to the limits of integration? As far as I understood, this is generally a hard problem, but I was wondering whether the specific structure of the inequalities can be exploited to simplify it.
Any hint is appreciated.