Let $z$ be the measured coordinates of a point on a circle $c$ with center $x$ and radius $r$. Assume the probability of measuring $z$ given the circle $c$ is normally distributed by the distance between $z$ and the border of $c$: \begin{equation} p(z|c) = \mathcal{N}(|z-x|; r, \sigma^2) \end{equation} We want to integrate the probability over all possible circles that are within a specified annulus $A$ with inner radius $r_{\min}$ and outer radius $r_{\max}$ and the center at the origin. Given a radius $r$, the set of all circles with radius $r$ within annulus $A$ is the set of circles for which the distance between the center and origin is at most: \begin{equation} \min(r-r_{\min},~ r_{\max}-r) \end{equation}
We describe the coordinates of the circle's center $x$ and the measurement $z$ by polar coordinates $(r_x, \theta_x)$ and $(r_z, \theta_z)$ respectively. Without loss of generality we assume that the polar coordinate system is orientated such that $\theta_z = 0$. By using the law of cosines the distance between the circle's center and $z$ is: \begin{equation} |z - x| = \sqrt{r_z^2 + r_x^2 - 2r_z r_x \cos \theta_x} \end{equation} The integral over all possible circles in annulus $A$ then becomes: \begin{align} & \int_{r_{\min}}^{r_{\max}}{ \int_{0}^{2\pi}{ \int_{0}^{\min(r-r_{\min},~ r_{\max}-r)} { p(z | x) dr_x} d\theta_x } dr} \\ & = \int_{r_{\min}}^{\tfrac{r_{\min}+r_{\max}}{2}}{ \int_{0}^{2\pi}{ \int_{0}^{r - r_{\min}} { p(z | x) dr_x} d\theta_x } dr} + \int_{\tfrac{r_{\min}+r_{\max}}{2}}^{r_{\max}}{ \int_{0}^{2\pi}{ \int_{0}^{r_{\max} - r} { p(z | x) dr_x} d\theta_x } dr} \end{align} with \begin{equation} p(z | x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\tfrac{\left(\sqrt{r_z^2 + r_x^2 - 2r_z r_x \cos \theta_x} - r\right)^2}{2\sigma^2}} \end{equation} Now my question is how to compute or to approximate integrals of the form \begin{equation} \int_{r_a}^{r_b}{ \int_{0}^{2\pi}{ \int_{0}^{r_c} { \frac{1}{\sigma \sqrt{2\pi}} e^{-\tfrac{\left(\sqrt{r_z^2 + r_x^2 - 2r_z r_x \cos \theta_x} - r\right)^2}{2\sigma^2}} dr_x} d\theta_x } dr} \end{equation}