It is well known that the discretization of a finite element method results in a linear system of equations, in which the matrix and the load vector are usually computed by numerical integration. The inexact integration may lead to a different linear system, and consequently, produce a different finite element solution. In this talk, we will first discuss the existing results on the impact of quadrature rules on the finite element approximation in the energy norm. Then, a sharp estimate on the convergence rate of the finite element approximation with numerical integration for linear functionals will be presented. This is the joint work with Ivo Babuska and Uday Banerjee.