A reflection intensity integration method is presented based uponab initiocalculation of three-dimensional (x, y, ω) reflection boundaries from a few physical crystal and instrument parameters. It is especially useful in challenging circumstances, such as the case of a crystal that is far from spherical, anisotropic mosaicity, α1α2peak splitting, interference from close neighbours, twin lattices or satellite reflections, and the case of streaks from modulated structures, all of which may frustrate the customary profile-learning and -fitting procedures. The method, calledEVAL-14, has been implemented and extensively tested on a Bruker Nonius KappaCCD diffractometer.