A third order accurate corrected trapezoidal is developed to integrate numerically the singular kernels of Laplace and Stokes equations over a class of parameterizable surfaces with special focus on cylindrical surfaces. The corrected trapezoidal rule has so far been applied to flat surfaces on equidistant grids for the kernels of Laplace and Stokes. Corrected trapezoidal rules are based on the standard trapezoidal rule where the singular point, which is assumed to be a discretization point, is omitted. To account for the omitted point corrected weights are computed which are applied locally in a vicinity of, and at, the singular point. For general surfaces the weights depend on the position of the singular point on the surface leading to specific weights for each grid point. However we identify a special class of manifolds for which universal weights, independent of the position relative to the surface, can be computed. We select from this class of surfaces the model problem of a cylinder for which we explicitly develop and validate quadrature rules for both the kernel of Stokes and Laplace equations. This quadrature rule can be applied to simulations of pipe flows in conjunction with e.g. particle suspensions, fiber suspensions, swimming micro-organisms. Here we validate the obtained quadrature rule by computing the drag on a spheroidal particle positioned on the inner axis of a cylinder.