Summary: Error estimates for midpoint, trapezoid, Simpson’s, Maclaurin’s, 3/8-Simpson’s and Boole’s type rules are obtained. Some related inequalities of Ostrowski’s type are pointed out. These results are obtained for mappings of bounded variation, Lipschitzian, and absolutely continuous mappings whose first derivatives are belong to \(L_p[a,b]\) (\(1\leq p\leq\infty\)). Applications to numerical integration are provided.