# Surface integral of spherical shell in a $e^{-x^2}$ field

I have a density field which is spherically-symmetric. The field is centred at the origin and is parametrized by a radius $R$ and a falloff parameter $k$. If I sample a point whose position vector is $\overrightarrow{r}$ from the origin, the density $\rho$ is given by

$$\rho(\overrightarrow{r})=e^{-k(|\overrightarrow{r}|-R)^2}$$

I want to sample this field by integrating over the surface of another sphere. This centre of this sampling sphere has position vector $\overrightarrow{S}$ and radius $R_S$.

I know a bit about calculus but not surface integrals. I want to integrate $\rho(\overrightarrow{r})$ over all points on the sample sphere, i.e. all points such that $|\overrightarrow{r}-\overrightarrow{S}|=R_S$.

How do I do this?