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We analyze the error in typical moment method scattering solutions for smooth cylindrical geometries and TM-polarized fields. We consider the magnetic field integral equation (MFIE), electric field integral equation (EFIE), and combined field integral equation. To quantify the impact of mesh element size, approximate integration of moment matrix elements, and geometrical discretization error on the accuracy of computed surface currents and scattering amplitudes, we derive error estimates analytically for the circular cylinder. For pulse functions and point testing, current and scattering amplitude errors are generally second order in the mesh element width. The convergence rate worsens to first order if low-order numerical quadrature is used to evaluate moment matrix elements with the EFIE formulation, or if a flat-facet mesh is employed with the MFIE formulation. Third-order convergence is also possible in some special cases. These results for the circular cylinder are empirically compared to computed error values for other smooth scatterer geometries. Convergence rates with respect to mesh refinement agree, but scatterers with regions of high curvature exhibit increased absolute error.