The Shifrin perturbation theory is applied to the scattering of light from finite cylinders and spheroids, and a procedure is described for averaging over the possible orientations of the symmetry axes of these various target shapes. The possible axis orientations of very long cylinders is limited by the conical shape of the scattering patterns, and the manner of taking this into account in the averaging process is detailed. It is found that the scattering intensities for the scattering of light from randomly oriented long cylinders, with aspect ratios (length/diameter) less than about 200, show differences compared with infinite cylinders of the same radius and dielectric constant (m=1.5). This is especially true for high and low scattering angles and when the plane of linear polarization is changed by the scattering. Moreover, the scattering intensities for the scattering of light from randomly oriented short cylinders and spheroids, with aspect ratios ranging from 0.1 to 10 and the same volume, show significant differences when the plane of polarization is changed by the scattering.