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