The area under a curve specified by measured values

The problem is addressed of determining numerical approximations to the area under a curve specified by arbitrarily spaced data. A formulation of this problem is given in which the data are used to model the curve as a piecewise polynomial, each piece having the same degree. That piecewise function is integrated to provide an approximation to the area. A corresponding compound quadrature rule is derived. Different degrees of polynomial give rise to different orders of quadrature rule. The widely used trapezoidal rule is a special case, as is the Gill–Miller rule. The remainder of the paper is concerned with evaluating the measurement uncertainties associated with the approximations to the area obtained by the use of these rules in the case where the data ordinates correspond to measured values having stated associated uncertainties. The case in which there is correlation associated with the measured values, frequently arising when the measured values are obtained using the same measuring instrument, is also treated. A statistical test is used to select a suitable polynomial degree.