The Born approximation is used to generate a sequence of iterates approximating spectral components of the continuous spectrum associated with longitudinally-uniform dielectric waveguides. Quantification of the electromagnetic fields within these structures is provided by an axial Fourier transform-domain electric field integral equation (EFIE) for the transverse components. Successive approximations of solutions to this EFIE for practical dielectric waveguiding structures are obtained. Convergence criteria of the Neumann series along with error estimates for each iterate are examined.