Integrals and Areas Expressed as Bilinear Forms of Discrete Defining Data

An ordered set of discrete data-pairs (x_k, y_k) is supposed givenfor k = 1(1)n, derived perhaps as a consequence of experiment,and indicating some form of relationship between x and y. Itis required that these data be interpolated in some systematicway so as to establish this relationship in the form of a rectifiablecurve in the (x, y) plane, and that y is then to be integratedwith respect to x (or vice versa), thereby defining an areain this plane. We shall here consider interpolation schemeswhere by this integral can be expressed as a bilinear form _jKC_jkY_jx_k,the "weighting" coefficients C_jk being numbers independent ofthe data values (though not necessarily of their number, n). The expressions obtained sepcialize to Gregory formulae, andti Simpson's Rule and other forms of Cotes formulae in suitablecontexts, but the information is primarily of use in dealingwith unequally-spaced data values, or in estumating the areawithin a closed curve. The simplicity of the numerical algorithmis clearly unaffected by the geometric complexity of the interpolatedcurve. Values for the weighting coefficients are proposed, and theresults of a numerical experiment are described which teststhe applicability of the formulae described.