Trapezoidal Formula for Definite Integrals

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Theorem

Let $f$ be a real function which is integrable on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}\right\}$ form a normal subdivision of $\left[{a \,.\,.\, b}\right]$:

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


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

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

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