Truncation Error from Trapezoid Rule

Theorem

Let $f: \R \to \R$ be a real function.

Let the trapezoid rule be used to integrate $f$ over the closed interval $\closedint a b$ using $n$ subintervals of length $h$.

Then the truncation error $E$ is given by:

$E = \dfrac {-h^2 \paren {b - a} \map {f} \alpha} {12}$

where the second derivative $f$ is evaluated at some unspecified point $\alpha \in \closedint a b$.

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Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): error: 1. (in numerical computation) Truncation errors
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error: 1. (in numerical computation) Truncation errors