Notes on two definite quadrature formulae of order three

In two recent papers from 2017 and 2018 two definite quadrature formulae of order three were obtained which are modifications of the compound trapezium and midpoint quadrature rules, respectively. The criteria applied to the construction of these definite quadrature formulae is minimization of their error constants through the usage of appropriate formulae for numerical differentiation. In this note, we show that the aforementioned definite quadrature formulae also meet another criteria for optimality. For integrands having a continuous third derivative with a permanent sign, we present another error estimates, which are easier to evaluate and in many cases provide better error bounds.