Suppose I have a non-negative function $f:\mathbb{R}^N \to [0, +\infty)$ that defines two different (unnormalized) probability densities on two separate subsets $A, B \subset \mathbb{R}^N$ with $A \cap B = \emptyset$ and, importantly, $A$ and $B$ are both compact, e.g. $A = [-1, 1]$ and $B= [-2, 2]$.
$$ \widetilde{p}(x) = \begin{cases} f(x) & \text{if } x\in A \\ 0 & \text{if } x \notin A\end{cases} \\ \widetilde{q}(x) = \begin{cases} f(x) & \text{if } x\in B \\ 0 & \text{if } x\notin B\end{cases} $$ Now suppose a set of samples from each $$ x_1^p,\ldots, x_N^p \sim \widetilde{p}(x) \\ x_1^q, \ldots, x_N^q \sim \widetilde{q}(x) $$ How can I use these samples to approximate the ratio of normalizing constants $$ \frac{Z_q}{Z_p} = \frac{\displaystyle \int_B \widetilde{q}(x)dx}{\displaystyle \int_A \widetilde{p}(x)dx} $$