to g (in the classical sense) if Examples and special cases Differentiable g(x) Given a which is continuously differentiable over it can be shown that there is the equality where the integral on the right-hand side is the standard Riemann integral, assuming that can be integrated by the Riemann–Stieltjes integral. More generally, the Riemann integral equals the Riemann–Stieltjes integral if is the Lebesgue integral of its derivative; in this case is said to be absolutely continuous. It may be the case that has jump discontinuities, or may have derivative zero almost everywhere while still being continuous and increasing (for example, could be the Cantor function or “Devil's staircase”), in either of which cases the Riemann–Stieltjes integral is not captured by any expression involving derivatives of g. Riemann integral The standard Riemann integral is a special case of the Riemann–Stieltjes integral where . Rectifier Consider the function used in the study of neural networks, called a rectified linear unit (ReLU). Then the Riemann–Stieltjes can be evaluated as where the integral on the right-hand side is the standard Riemann integral. Cavalieri integration Cavalieri's principle can be used to calculate areas bounded by curves using Riemann–Stieltjes integrals. The integration strips of Riemann integration are replaced with strips that are non-rectangular in shape. The method is to transform a "Cavaliere region" with a transformation , or to use as integrand. For a given function on an interval , a "translational function" must intersect exactly once for any shift in the interval. A "Cavaliere region" is then bounded by , the -axis, and . The area of the region is then where and are the -values where and intersect . Notes References via HathiTrust Definitions of mathematical integration Bernhard Riemann