Error bounds for the trapezoidal quadrature rule and Simpson’s method are derived under the various (typically weak) smoothness assumptions on the integrand function. Some of the results are closely related to those of P. Köhler
[Math. Res. 86, 141-150 (1995; Zbl 0843.41011
), and estimates for linear remainder functionals by the modulus of continuity, in B. Bojanov (ed.), Open problems in approximation theory, SCT Publ. Comp., Singapore, 109-124 (1994).