Trapezoidal Rule: Discrete Integration via Linear Interpolation — E8 Intelligence Research

FINDING: Trapezoidal integration is a discrete numerical method for approximating definite integrals by summing areas of linear-interpolated segments between data points. | MATH: For discrete points \\((x_i, y_i)\\), the rule is \\(\\int_{x_0}^{x_n} f(x)\\,dx \\approx \\sum_{i=0}^{n-1} \\frac{(x_{i+1}-x_i)(y_i+y_{i+1})}{2}\\). Error term: \\(-\\frac{(b-a)^3}{12n^2} f''(\\xi)\\), \\(\\xi \\in (a,b)\\). For uniform step \\(h\\), the sum simplifies to \\(\\frac{h}{2}\\left(y_0 + 2\\sum_{i=1}^{n-1} y_i + y_n\\right)\\). | CONNECTION: The factor \\(\\frac{1}{2}\\) in the trapezoid area is the arithmetic mean of the two endpoint heights — this is the discrete analogue of the continuous mean value theorem. The error term's \\(1/12\\) coefficient is notable: \\(1/12 = 0.08333...\\), and \\(12\\) is the order of the icosahedral rotation group (crystallographic point group \\(I_h\\) has 120 elements, but the rotational subgroup has 60; the tetrahedral group has 12). The \\(1/12\\) also appears in the Euler–Maclaurin correction, link Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com