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We introduce a new approach for combining the integral equation and high frequency asymptotic techniques, e.g., the geometrical theory of diffraction. The method takes advantage of the fact that the Fourier transform of the unknown surface current distribution is proportional to the scattered far-field. A number of asymptotic methods are currently available that provide good approximation to this farfield in a convenient analytic form which is useful for deriving an initial estimate of the Fourier transform of the current distribution. An iterative scheme is developed for systematically improving the initial form of the high frequency asymptotic solution by manipulating the integral equation in the Fourier transform domain. A salient feature of the method is that it provides a convenient validity check of the solution for the surface current distribution by verifying that the scattered field it radiates indeed satisfies the boundary conditions at the surface of the scatterer. Another important feature of the method is that it yields both the induced surface current density and the far-field. Diffraction by a strip (two-dimensional problem) and diffraction by a thin plate (three-dimensional problem) are presented as illustrative examples that demonstrate the usefulness of the approach for handling a variety of electromagnetic scattering problems in the resonance region and above.