Trapezoidal Formula for Definite Integrals

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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:

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

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


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Proof


This theorem requires a proof.
In particular: Graphical approach based on dividing the area under the curve as trapezia. We need to explain rigorously what an "approximation" means, and we may also want to quantify the error.
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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

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