Expected value of posterior vs. success probability

Context

Suppose I have two models, $H_1$ and $H_2$ for which I know the prior probabilities $p(H_1)$ and $p(H_2)$. Furthermore, I know the class-conditional distributions $p(x|H_1)$ and $p(x|H_2)$ of a random variable $X \in \mathbb{R}^n$. I get to observe a realization of $X$, call it $x_0$.

The posterior distribution of the models given the observation I just made can be obtained via Bayes' formula:

$$p(H_i|x_0) = \frac{p(x_0|H_i)p(H_i)}{p(x_0)}$$

And the Bayes-optimal decision rule is the (very intuitive) MAP rule:

$$\hat{H} = \operatorname*{arg\,max}_{H_i} p(H_i|x_0)$$

Now, let's say I'm interested in how certain I am when I'm deciding for $H_1$. The probability that I'm correctly identifying $H_1$ when indeed $X$ is generated from model $H_1$ is:

$$P_{suc,H_1} = \int_{R_1}p(x|H_1)dx,$$

where $R_1$ is the decision region associated to $H_1$, i.e.

$$R_1 = \{x \in \mathbb{R}^n : \operatorname*{arg\,max}_{H_i} p(H_i|x_0) = H_1\}$$

Problem

Now that I've introduced the terminology & notation, let me come to the actual question ;)

In my case, $R_1$ has a complicated shape (it's a pretty much arbitrary convex polytope), and I'm just not able to perform the integration in order to obtain $P_{suc,H_1}$. I'm looking for alternative ways to get this quantity.

My intuition is as follows: the probability of success is related (equal?) to the expected value of the posterior under the hypothesis $H_1$. Mathematically:

$$P_{suc, H_1} \longleftrightarrow \mathbb{E}_{p(x|H_1)}[p(H_1|x)]$$

or:

$$\int_{R_1}1p(x|H_1)dx \longleftrightarrow \int_{\mathbb{R}^n}p(H_1|x)p(x|H_1)dx$$

Actually, I've already managed to prove that the two are not equal (in the left integral, replace $1$ with $p(H_1|x) + p(H_2|x)$; in the right integral, use $\mathbb{R}^n = R_1 + R_2$; some terms cancel out and you can analyze what's left over.)

Questions

I guess I really have two questions:

1. What's the interpretation of the quantity $\int_{\mathbb{R}^n}p(H_1|x)p(x|H_1)dx$? Intuitively, it should relate to $P_{suc,1}$, but I'm not sure how / why.
2. Is there any other way I could compute the error probability by integrations over some simpler regions (even if the integrand becomes more complicated)?

To answer your first question, see that \begin{align} P_{suc,H_1} &= \int_{R_1} p(x|H_1) \text{d}x = \int \mathbb{1}_{\{ x \in R_1 \}} p(x|H_1) \text{d}x = \mathbb{E}_{x \sim H_1} \left[\mathbb{1}_{\{ x \in R_1 \}} \right] \end{align} in other words, if you were to vary $x$ according to $H_1$, what would the expectation be of the random variable $\mathbb{1}_{\{ x \in R_1 \}}$. This is different from $\mathbb{E}_{x \sim H_1}\left[ p(H_1 |x) \right]$ which asks a similar question but replacing $\mathbb{1}_{\{ x \in R_1 \}}$ with $p(H_1|x)$. The first random variable is a on-off quantity while the latter is a typically smooth function.
To answer your second question, if you willing to achieve an approximate answer for $P_{suc,H_1} = \mathbb{E}_{x \sim H_1} \left[\mathbb{1}_{\{ x \in R_1 \}} \right]$ - try random sampling $x \sim H_1$, computing $\mathbb{1}_{\{ x \in R_1 \}}$ for each sample, and take the sample mean.