Suppose that $f(r,\theta)$ be a 2D probability density function in polar coordinates. How can I estimate the following integral $$I= \int_{\theta_1}^{\theta_2} f(r_0, \theta)g(r_0,\theta) d\theta$$ Using $N$ samples of the 2D distribution $(r_1,\theta_1),\ldots, (r_N,\theta_N) $?
In the above integration $g(r,\theta)$ is a given function and $r_0$ is a constant.
My thoughts:
If I had a 1D PDF, $f(\theta)$, and samples in the form of $(\theta_1),\ldots, (\theta_N) $, I could use the Monte Carlo estimation:
$$ \int f(\theta)g(\theta) d\theta\approx \frac{1}{N}\sum_{i=1}^N g(\theta_i)$$
Also, if I wanted to estimate a single point of the PDF such as $f(r_0, \theta_0)$ I could use the k-Nearest Neighbor (kNN) method to estimate the density as:
$$f(r_0, \theta_0)\approx \frac{k}{N \pi R_{(k)}^2(r_0, \theta_0)}$$
where $R_{(k)}(r_0, \theta_0)$ is the distance of $k$th nearest neighbor (from samples) of the point $(r_0, \theta_0)$ with respect to this point.
I guess the solution might be a combination of the two concepts. I have no idea on how to proceed.