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 $f$ with respect to $x$ from $a$ to $b$ can be approximated as:

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

where:

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

$m = \dfrac n 2$

Proof

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Also known as

Simpson's Rule is also known as the Parabolic Formula.

It can also be seen as Simpson's Formula, but this may be confused with Simpson's Formulas, which is a set of completely different results.

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: $15.17$