Let $\Delta^n=\{x\in[0,1]^n\colon \sum_{i=1}^nx_i\le 1\}$ be a simplex.
I want to compute $\int_{\Delta^n}\exp\left (\sum_{i=1}^nx_i\right )\,\mathrm{d}\lambda(x)$.
As I have already determined the volume of the simplex, I tried a similar approach for the integral.
$\int_0^1...\int_0^{1-x_1-x_2...-x_{n-1}}\exp\left ( \sum_{i=1}^nx_i\right ) \mathrm{d}x_n...\mathrm{d}x_1$
But if I compute this integral it might not be positive (depending on $n$), so this cannot be correct.