Newton's Three-Eighths Rule

Theorem

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

Let $P = \set {a = x_0, x_1, x_2, \ldots, x_{3 n - 1}, x_{3 n} = b}$ form a normal subdivision of $\closedint a b$:

$\forall r \in \set {1, 2, \ldots, 3 n}: x_r - x_{r - 1} = \dfrac {b - a} {3 n}$

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

$\ds \int_a^b \map f x \rd x$

$\approx$

$\ds \dfrac {3 h} 8 \paren {\map f {x_0} + 3 \sum_{r \mathop = 1}^n \paren {\map f {x_{3 i - 2} } + \map f {x_{3 i - 1} } } + 2 \sum_{r \mathop = 1}^{n - 1} \map f {x_{3 i} } + \map f {x_{3 n} } }$

$\ds $

$=$

$\ds \dfrac {3 h} 8 \paren {\map f {x_0} + 3 \map f {x_1} + 3 \map f {x_2} + 2 \map f {x_3} + 3 \map f {x_4} + 3 \map f {x_5} + 2 \map f {x_6} + \cdots + \map f {x_{3 n} } }$

where:

$h = \dfrac {b - a} {3 n}$

Proof

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Also known as

is also known as Newton's rule.

However, there are many results with Isaac Newton's name on them, so to reduce confusion and ambiguity, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the full version of the name.

Source of Name

This entry was named for Isaac Newton.

Sources

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Newton's rule
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Newton's rule