I want to evaluate $I = \int_V f(\vec{x}) d\vec{x}$. The classical Monte Carlo method is to sample uniformly from within the integration volume $V$, and then compute $I \approx V \frac{1}{N} \sum_{i=1}^{N} f(\vec{x})$.

What if I sampled the $\vec{x}$'s using a MCMC approach (e.g. Metropolis-Hastings, slice sampling), how do I compute $I$? Specifically, if we use the above formular, what is $V$?