= Solution
The incident <plane wave> carries <probability current> $j_{\rm in}=\hbar k/m$. For the separated outgoing wave from the previous part,
$$
j_r^{\rm sc}=\frac\hbar m\operatorname{Im}(\psi_{\rm sc}^*\partial_r\psi_{\rm sc})
\sim\frac\hbar{mr}|f(\theta)|^2.
$$
The rate through a distant arc is $rj_r^{\rm sc}\,d\theta$. Dividing by incident flux defines the differential two-dimensional <scattering cross-section>,
$$
\boxed{\frac{d\lambda}{d\theta}=\frac{|f(\theta)|^2}{k},\qquad
\lambda=\frac1k\int_0^{2\pi}|f(\theta)|^2\,d\theta.}
$$
Because $f$ is <dimensionless>, $\lambda$ has units of length, as appropriate for scattering in a plane.
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