Simpson's rule

Summary

Simpson's rule approximates $\int_a^b f$ by integrating the parabola through the values of $f$ at $a$, at the midpoint, and at $b$, which gives $\frac{b - a}{6}\big(f(a) + 4 f(m) + f(b)\big)$. It is the closed Newton–Cotes formula with three nodes, and it is exact for all cubic polynomials. Its composite form has error of order $h^4$ for smooth integrands, which makes it a standard simple method.