Trapezium rule

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Trapezoidal_rule_illustration.png
The function f(x) (in blue) is approximated by a linear function (in red).

In mathematics, the trapezium rule (the British term) or trapezoidal rule (the American term) is a way to approximately calculate the definite integral

[itex] \int_{a}^{b} f(x)dx.[itex]

The trapezium rule works by approximating the region under the graph of the function [itex]f(x)[itex] by a trapezium and calculating its area. It follows that

[itex] \int_{a}^{b} f(x) dx \approx (b-a)\frac{f(a) + f(b)}{2}.[itex]

To calculate this integral more accurately, one first splits the interval of integration [itex][a, b][itex] into n smaller subintervals, and then applies the trapezium rule on each of them. One obtains the composite trapezium rule:

[itex]\int_a^b f(x)\,dx \approx \frac{b-a}{n} \left( {f(a) + f(b) \over 2} + \sum_{k=1}^{n-1} f \left( a+k \frac{b-a}{n} \right) \right).[itex]

This can alternatively be written as:

[itex]\int_a^b f(x)\,dx \approx \frac{b-a}{2n} \left(f(x_0) + f(x_n) + 2f(x_1) + 2f(x_2)+\dots+2f(x_{n-1}) \right).[itex]

The trapezium rule is one of a family of formulas for numerical integration called Newton-Cotes formulas. Simpson's rule is another, often more accurate, member of the same family. Simpson's rule and other like methods can be expected to improve on the trapezium rule for functions which are twice continuously differentiable; however for rougher functions the trapezium rule is likely to prove preferable. Moreover, the trapezium rule tends to become extremely accurate when periodic functions are integrated over their periods, a fact best understood in connection with the Euler-Maclaurin summation formula.

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