A method for estimating upper bounds for errors in a general type of quadrature rule is described. By using asymptotic expressions for the integrand in the contour integral form of the error term, an inequality is obtained in the form of a product of two terms, one dependent only on the quadrature rule and the other dependent on the function to be integrated. This product depends on a parameter which may be varied to find a least upper bound from a selection of values of the parameter. Tables of the quadrature-dependent term from which the bounds are readily obtained are given for few of the Gauss–Legendre, Newton–Cotes and Gauss–Laguerre formulas. An alternative approach which requires much more analysis is given for the Gauss–Laguerre formula.