Simpson's Rule

Let $h\equiv (b-a)/n$, and assume a function $f(x)$ is defined at points $f(a+kh)=y_k$ for $k=0$, ..., $n$. Then

$$\int_a^b f(x)\,dx = {\textstyle{1\over 3}} h (y_1+4y_2+2y_3+4y_4+\ldots+2y_{n-2}+4y_{n-1}+y_n)-R_n,$$

where the remainder is

$$R_n={\textstyle{1\over 90}} (b-a)^4 f^{(4)}(x^*)$$

for some $x^*\in [a,b]$.

See also Bode's Rule, Newton-Cotes Formulas, Simpson's 3/8 Rule, Trapezoidal Rule

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 886, 1972.