Quadrature methods for approximating the definite integral of a function f(t) over an interval [a,b] are in common use. Examples of such methods are the Newton-Cotes formulas (midpoint, trapezoidal and Simpson methods etc.) and the Gauss-Legendre quadrature rules, to name two types of quadrature. Error bounds for these approximations involve higher order derivatives. For the Simpson method, in particular, the error bound involves a fourth-order derivative. Discounting the fact that calculating a fourth-order derivative requires a lot of differentiation, the main concern is that an error bound for the Simpson method, for example, is only relevant for a function that is four times differentiable, a rather stringent condition. This paper caters for functions for which derivatives exist only of order lower than normally required. A number of quadrature methods are considered and error bounds derived involving only lower order derivatives that can be used depending on the smoothness of the function. (Contains 2 tables and 6 figures.)