# Trapezoidal Formula for Definite Integrals

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## Contents

## 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 $\closedint a b$:

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

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

- $\displaystyle \int_a^b \map f x \rd x \approx \dfrac h 2 \paren {\map f {x_0} + \map f {x_n} + \sum_{r \mathop = 1}^{n - 1} 2 \map f {x_r} }$

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

## Proof

## Also known as

Non-US sources refer to this rule as the **trapezium rule**, as a result of the fact that the terms **trapezoid** and **trapezium** have the opposite definitions outside the US.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 15$: Approximate Formulas for Definite Integrals: $15.16$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**trapezoidal rule** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**trapezoidal rule (trapezium rule, trapezoid rule)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**trapezium rule, trapezoidal rule**