  A recently re-discovered variant of the Backus-Gilbert algorithm for spectral
reconstruction enables the controlled determination of smeared spectral
densities from lattice field theory correlation functions. A particular
advantage of this approach is the \emph{a priori} specification of the kernel
with which the underlying spectral density is smeared, allowing for variation
of its peak position, smearing width, and functional form. If the unsmeared
spectral density is sufficiently smooth in the neighborhood of a particular
energy, it can be obtained from an extrapolation to zero smearing-kernel width
at fixed peak position. A natural application for this approach is scattering
processes summed over all hadronic final states. As a proof-of-principle test,
an inclusive rate is computed in the two-dimensional O(3) sigma model from a
two-point correlation function of conserved currents. The results at finite and
zero smearing radius are in good agreement with the known analytic form up to
energies at which 40-particle states contribute, and are sensitive to the
4-particle contribution to the inclusive rate. The straight-forward adaptation
to compute the $R$-ratio in lattice QCD from two-point functions of the
electromagnetic current is briefly discussed.
