For a sharp notch with curved edges and subjected to surface tractions along the edges, the fracture parameters (in particular, the stress intensity factors and the size of a singular-dominant zone) are significantly affected by the near-tip geometric and loading conditions. In this paper, a pair of contour integrals termed is presented for calculating the mixed-mode stress intensity factors at such a sharp notch-tip. Furthermore, by proper use of the integrals, the extent of the singular-dominant zone can be effectively characterized. Since no a priori auxiliary (or, complementary) solutions are required in its formulation, the approach appears to be feasible for problems of arbitrary notch angles and curved shapes. Also, no special treatments are required for the modeling of the near-tip singular behavior so that the integration can be performed by direct use of numerical schemes such as finite element method.