Describing the region of integration $\mathbb{R}_+^n$ for arbitrary $n$

Question

Let $f_n:\mathbb{R}^n \to \mathbb{R}$. I would like to know an elegant way of writing the integral $$ I_n = \int_{\mathbb{R}^n_+} f_n(x)~\mathrm{d}x $$ in Mathematica.

For fixed $n$, the best I could do was for the region $\mathbb{R}^n$ as the following code:

x = {x1,x2,x3,x4,x5,x6,x7,x8,x9};
Integrate[f[n,x], x \[Element] FullRegion[n]]

More specifically, I would like to vary $n$. Then, considering a minimal working example with $f_n(x) = \exp\left(-\sqrt{\sum_{k=1}^n x_k}\right)$ and using the HeavisideTheta function to get the positive reals, I ended up with

x = {x1, x2, x3, x4, x5, x6, x7, x8, x9};
f[n_, x_] := Exp[-Sqrt[Sum[x[[k]], {k, 1, n}]]];
int[n_] := 
  NIntegrate[Product[HeavisideTheta[x[[k]]], {k, 1, n}] f[n, x], 
   Table[x[[k]], {k, 1, n}] \[Element] FullRegion[n]];

Among the improvements, I would like to avoid the HeavisideTheta trick and defining the x={x1,x2,...,xn}.

Answer 1

Instead of using "list" variables, like x = {x1, x2, ...}, use "indexed" variables, x[1], x[2], ..., which can be easily generated with Array. Positive region can be defined, for example, via ImplicitRegion.

Clear["Global`*"];

positiveRegion[x_] := ImplicitRegion[x >= 0, x]

f[x_] := Exp[-Sqrt[Total[x]]];

int[n_] := With[{x = Array[x, n]}, Integrate[f[x], x ∈ positiveRegion[x]]];

Table[{n, int[n]}, {n, 5}]
(* {{1, 2}, {2, 12}, {3, 120}, {4, 1680}, {5, 30240}} *)