Quadrature over the sphere

Consider integration over the unit sphere in R^3, especially when the integrand has singular behaviour in a polar region. In an earlier paper, a numerical integration method was proposed that uses a transformation that leads to an integration problem over the unit sphere with an integrand that is much smoother in the polar regions of the sphere. The transformation uses a grading parameter "q". The trapezoidal rule is applied to the spherical coordinates representation of the transformed problem. In this paper, we extend those results and also examine superconvergence.