Trapezoidal Formula for Definite Integrals

Theorem
Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.

Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$:
 * $\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$

Then the definite integral of $f$ $x$ from $a$ to $b$ can be approximated as:


 * $\ds \int_a^b \map f x \rd x \approx \dfrac h 2 \paren {\map f {x_0} + \map f {x_n} + \sum_{r \mathop = 1}^{n - 1} 2 \map f {x_r} }$

where $h = \dfrac {b - a} n$.

Also known as
Non-US sources refer to this rule as the trapezium rule, as a result of the fact that the terms trapezoid and trapezium have the opposite definitions outside the US.