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Electromagnetic and Acoustic Wave Scattering, Method of Moments and Error Bound studies examples of transverse magnetic (TM) and transverse Electric (TE) electromagnetic scattering, and acoustic scattering by a sound soft surface leading to Fredholm integral equations of the first kind. In each case the scattering problem is solved by the Galerkin method (method of moments with identical sets of basis and testing functions) and an error bound is derived for the approximation. Error bounds are expressed with norms in Sobolev spaces of order 1/2 and 1. Bounds are also derived using values of the residuals solely on the strip or plate where they are defined. A variational method derives novel approximate solutions of the above scattering problems that minimize the error bounds. Approximate solutions, residuals and error bounds are illustrated in many color plots. Detailed algorithms are included in appendices with verified MATLAB code to compute many special functions such as Struve functions and successive primitives of Bessel and Hankel functions. MATLAB code is also included for matrices used in the method of moment with piecewise constant and piecewise linear functions.