In this work, the frequency domain current distributions induced on infinitely long cylinders of arbitrary cross section illuminated by transverse magnetic plane wave are obtained using the finite integral technique (FIT). These bodies may be composed of conducting, dielectric, or inhomogeneous materials. In the FIT, the Maxwell's equations in the integral form are solved directly, thus generating a sparse system matrix. The problem region is subdivided into triangular elements, and distinct constitutive parameters may be assigned to each element, thus modeling the inhomogeneous materials accurately and efficiently. Next, the measured equation of invariance (MEI) concept is used to truncate the mesh which preserves the sparsity of the system matrix. Further, MEI allows the termination of the grid close to the structure, in fact, only two layers, increasing the computational efficiency of the method. Several numerical examples are presented to illustrate the applicability of the method.