A more microscopic view of the cross section is

$\displaystyle \frac{dN}{dS} \sigma \rho dt dA= dn$,

where $N$ is the number of incident particle, $S$ is the area of the beam, $\sigma$ is the total cross section, $\rho$ is the particle density in the target, $A$ is the area of the target, $t$ is the thickness of the target, and $n$ is the particle detected.

The cross section is a constant, so, after integration on the target area and thickness,

$\displaystyle \sigma = \frac{n}{N \rho t}$.

If the beam is not uniform, $dN/dS = f(S)$ is a function of $S$. The integration has to be careful.