Error bounds for approximations of the definite integrals (Q1611461)

This paper is concerned with the approximation of the Riemann integral \(\int _a^bf(x) dx \) of a differentiable function \(f\) by the classical trapezoidal, midpoint and Simpson rules. The authors show that the errors are of first order accuracy and depend only on \(||f^\prime ||\). Bounds for the error terms are also given for a convex function \(f\) in the case that both \(m=\int _+^\prime f(a)\) and \(M=\int _-^\prime f(b)\) exist.