or upper Darboux sum, n. the weighted sum of the products of the supremal values of a function on a succession of subintervals of a given interval with the lengths of the subintervals; hence, the area under the step function of which the value is the supremum of the given function on each subinterval, as shown below. The limit of this sum of products, as the lengths of the subintervals tend to zero, is the upper integral of the function. Compare lower sum. See Riemann integral.