On the remainder form of certain formulas of mechanical quadrature

Abstract

The class of formulas considered in this paper belong to a type where the required integral is approximately represented by a linear function of a certain number of equidistant values of the integrand. Formulas of this type may, for instance, be obtained by integrating Lagrange's interpolation formula between finite limits, as is well known. As regards the function to be integrated we assume only, that it possesses, throughout the interval of integration, a continuous differential coefficient of the highest order of which use is made in deriving the particular formula under consideration. It is not necessary, then, that the function should possess differential coefficients of every order, much less, that it should be a polynomial.