The Generalized Method of Exhaustion

The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of exhaustion) and an independent algebraic proof are provided. Applications provided as examples include use of the formula to generate new series for common functions as well as computing the group velocity distribution resulting from waves diffracted from an aperture.


Introduction
The method of exhaustion was used by the ancient Greeks to integrate a limited number of functions before the development of calculus [1]. It will be shown here that this ancient method can be generalized to a simple formula to integrate functions under very general conditions, assuming that the integral exists over the interval in question. The formula is The proof is simple and geometric in nature, directly following the procedure originally used for the method of exhaustion. An independent algebraic proof is also provided for the special case for which a = 0.

Geometric Proof
The geometric proof is a generalization of the successive approximation method historically used to obtain the exact area under the simple curves previously integrated by this approach. Consider an arbitrary function f(x) that is piecewise continuous on [a,b]. The procedure to find the area under f(x) on [a,b] involves successive approximations with triangles that first intersect f(x) at the midpoint, x = a + (b -a)/2, and then at the new midpoints, x = a + (b -a)/4, x = a + 3(b -a)/4, and so on. The first such approximation is shown in figure  1. Its area is The second such approximation adds two new triangles (figure 2), each sharing two vertices with the original triangle, and each bisecting one of the remaining arcs of f(x) at its third vertex. The two triangles have the area A third such approximation (figure 3) adds the area Each new iteration n leads to 2 n -1 new triangles adding area A n to further refine the area estimate. This procedure is continued indefinitely to "exhaust" the remaining area, leading to the formula The special case for (5) when a = 0 is as follows The method of exhaustion will converge to the value of the integral at least as fast as a geometric series, because when each new triangle is small enough so that the local curvature between intersection points on f(x) is slowly varying, it will exhaust more than 1/2 of the remaining area left from the previous approximation.
There are an infinite number of possible variations on this procedure. For example, each new iteration may involve a different number of triangles to exhaust the remaining area, or each new triangle could intersect the function at a location other than the midpoint.

Algebraic Proof
An independent proof for (6) can be obtained starting with the identity [2] sin a a This expression can be rewritten as follows: Since sin ba it follows that Defining and noting that Setting Note that H(x) can be in the form of a Fourier series which sums to the following: so

Applications
The most immediate applications are series expressions for common functions that might otherwise be very difficult to derive. Some examples are presented below: In addition to the above expressions, this method can lead to new insights into certain physical problems. One example involves the diffraction of waves though a two-dimensional aperture in an infinite screen. This problem has been solved exactly in integral form [3][4][5] so that the field at every point depends on the Fourier transform F(k x ,k y ) of the aperture at z=0: Note that (26) satisfies the Helmholtz equation, everywhere (k = ω 0 /c) as well as the boundary conditions on the screen. Under suitable conditions, the extension of (5) to convergent improper integrals can be made as follows, i.e., Equation (28)  tends to a finite limit L as B→ ∞.
With regard to (26), however, it can be noted that propagating waves will only occur when the transverse wavenumber (k 2 x +k 2 y ) 1/2 is lower than the cutoff wavenumber, so in the far field the integral can be evaluated to ±k with good accuracy as follows (assuming that F(k x ,k y ) is an even function with respect to both k x and k y ): × cos(mxk/2 n ) cos(qyk/2 p )e izk √ 1−m 2 /4 n −q 2 /4 p It can be clearly seen from (30) that the resulting field is due to a summation of an infinite number of plane waves, each propagating at a different phase velocity (and hence a different group velocity) based on its value of k x and k y . Thus, the process of diffraction leads to a continuous distribution of group velocities, having an amplitude distribution governed by the Fourier transform of the aperture. This insight becomes particularly clear upon application of (5) to expand (26). Such a result has an effect on Doppler shifts both in acoustics and in electromagnetic wave propagation, effects that are not otherwise apparent without the use of the generalized method of exhaustion.