First, we present formal aspects of the scattering of plane elastic waves by a single elastic cylindrical inclusion or a cavity of an arbitrary cross section embedded in an isotropic homogeneous elastic medium of different properties. An integral equation is used to derive expressions for the displacement field valid in the entire domain. Thereafter the Born approximation for the solution is derived. Formulas for the scattered amplitudes and the scattering cross sections are presented. The validity of the approximation is examined by comparing the results with those of the simple bodies for which the exact solutions are available in the literature. Some new results for the corresponding three‐dimensional case are also presented.