On a Posteriori Error Bounds of Trapezoidal Rule

In this paper, we discuss the average case setting and the probabilistic setting of composite Trapezoidal rule assuming that the class of integrand on an interval [0], [1] is equipped with a variant of the r-fold wiener measure based on the a finite number of function values for numerical integration. Moreover, we compute a posteriori bounds on the error of Trapezoidal rule from a probabilistic point of view. This a new a posteriori error bound is better than a bound that is commonly used in practice.