Trapezoidal rule

Summary

The trapezoidal rule approximates $\int_a^b f$ by the area of the trapezoid under the chord from $(a, f(a))$ to $(b, f(b))$. Its composite form splits $[a, b]$ into $N$ equal parts and adds the trapezoids, with error of order $h^2$ for smooth $f$, where $h = (b - a)/N$. It is the closed Newton–Cotes formula with two nodes, and it is remarkably accurate for smooth periodic functions over a full period.