I want to approximate the integral:
$$ I = \int f(\boldsymbol{x})d\boldsymbol{x} $$
where $\boldsymbol{x}$ is $d$-dimensional. I have a set of non-equally spaced points $\boldsymbol{x}_1, \dots, \boldsymbol{x_n}$, and the corresponding values $f(\boldsymbol{x}_1), \dots, f(\boldsymbol{x_n}$), and I would like to use them to approximate the integral $I$. Actually the integral is over $R^d$, but I know that $ f(\boldsymbol{x})$ goes to zero quite rapidly outside my sample $\boldsymbol{x}_1, \dots, \boldsymbol{x_n}$.
My question is the following: are there rules to approximate the integral using the non-equally spaced points? I've found simple rules for the univariate case, but for multiple dimensions. Thanks
Note: the points $\boldsymbol{x}_1, \dots, \boldsymbol{x_n}$ are given and I can't choose them.