Analysis of integrated optical cylindrical microresonators involves the coupling between a straight waveguide and a bent waveguide. Our (2D) variant of coupled mode theory is based on analytically represented mode profiles. With the bend modes expressed in Cartesian coordinates, coupled mode equations can be derived in a classical way and solved by numerical integration. Proper manipulation of the propagation matrix leads to stable results even in parameter domains of compact and/or radiative structures, which seemed unsuitable for a perturbational approach due to oscillations of the matrix elements along the propagation. Comparisons with FDTD calculations show convincing agreement.