Is it possible to examine a 3D integral by using Gauss-Hermite quadrature type technique? I mean there might be an equation like this (with analogy to 1D Gauss-Hermite quadrature):
$\int_{-1}^{-1} \int_{-1}^{1} \int_{-1}^{1} f(x,y,z) dx dy dz = \sum_{i} \sum_{j}\sum_{k} w_{ijk} f(i,j,k)$
I appreciate your help and suggestions!
Just looking the boundaries of your integral, I wouldn't use Gauss-Hermite. I would use Gauss-Chebyshev or Gauss-Legendre depending on the function $f$. You should just apply the quadrature formula with the order of integration.