Error Analysis of Curvilinear Integrals

We consider the computation of the curvilinear integrals over a smooth orpiecewise smooth curve,where an explicitly differentiable parametrization is not availiable.A parametrization is replaced by a piecewise polynomial interpolant of it and a Gaussian inte-gration formula is used.On the basis of error analysis,the relation between the number ofGaussian integration nodes and interpolation nodes is set up.A numerical example is givenand the numerical results are consistent with the theoretical results, so the validity of thetheoretical results is confirmed.