The p-version of the finite element methods requires the exact calculation of the stiffness matrix by a special form of numerical integration. As an alternative to classical techniques that are based on Gauss quadrature, we propose to use low order methods combined with extrapolation. To this purpose we derive asymptotic expansions for basic integration methods on triangles. In contrast to conventional extrapolation methods for elliptic equations these results use only a local analysis and can thus be used on unstructured meshes. We present a complete analysis and examples with practical suggestions for extrapolation-based high order finite element methods.