Trapezoidal Rule

The 2-point Newton-Cotes Formula

$$\int_{x_1}^{x_2} f(x)\,dx = {\textstyle{1\over 2}}h(f_1+f_2)-{\textstyle{1\over 2}}h^3 f''(\xi),$$

where $f_i\equiv f(x_i)$, $h$ is the separation between the points, and $\xi$ is a point satisfying $x_1\leq\xi\leq x_2$. Picking $\xi$ to maximize $f''(\xi)$ gives an upper bound for the error in the trapezoidal approximation to the Integral.

See also Bode's Rule, Hardy's Rule, Newton-Cotes Formulas, Simpson's 3/8 Rule, Simpson's Rule, Weddle's 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. 885, 1972.