Three Point Quadrature Rules Involving, at Most, a First Derivative\ud

A unified treatment of three point quadrature rules is presented in which the classical rules of mid-point, trapezoidal and Simpson type are recaptured as particular cases. Riemann integrals are approximated for the derivative of the integrand belonging to a variety of norms. The Grüss inequality and a number of variants are also presented which provide a variety of inequalities that are suitatable for numerical implementation. Mappings that\ud are of bounded total variation, Lipschitzian and monotonic are also investigated with relation to Riemann-Stieltjes integrals. Explicit a priori bounds are provided allowing the determination of the partition required to achieve a\ud prescribed error tolerance.\ud It is demonstrated that with the above classes of functions, the average of a mid-point and trapezoidal type rule produces the best bounds