# Simpson's Rule

## Theorem

Let $f$ be a real function which is integrable on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}\right\}$ form a normal subdivision of $\left[{a \,.\,.\, b}\right]$:

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

where $n$ is even.

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

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

where:

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

## Proof

## Also known as

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

## Source of Name

This entry was named for Thomas Simpson.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 15$: Approximate Formulas for Definite Integrals: $15.17$