Simpson's Rule

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 $\closedint a b$:
 * $\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$

where $n$ is even.

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


 * $\displaystyle \int_a^b \map f x \rd x \approx \dfrac h 3 \paren {\map f {x_0} + \map f {x_n} + \sum_{r \mathop = 1}^{m - 1} 2 \map f {x_{2 m - 1} } + \sum_{r \mathop = 1}^{m - 1} 4 \map f {x_{2 m} } }$

where:
 * $h = \dfrac {b - a} n$
 * $m = \dfrac n 2$

Also known as
This rule is also known as Simpson's formula, or the parabolic formula.