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Trapezoidal rule
Numerical integration method / From Wikipedia, the free encyclopedia
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In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule)[lower-alpha 1] is a technique for numerical integration, i.e., approximating the definite integral:
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/4/40/Trapezoidal_rule_illustration.svg/320px-Trapezoidal_rule_illustration.svg.png)
The trapezoidal rule works by approximating the region under the graph of the function
as a trapezoid and calculating its area. It follows that
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/1/10/WikiTrap.gif/320px-WikiTrap.gif)
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite") trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule". Let be a partition of
such that
and
be the length of the
-th subinterval (that is,
), then
When the partition has a regular spacing, as is often the case, that is, when all the
have the same value
the formula can be simplified for calculation efficiency by factoring
out:.
The approximation becomes more accurate as the resolution of the partition increases (that is, for larger , all
decrease).
As discussed below, it is also possible to place error bounds on the accuracy of the value of a definite integral estimated using a trapezoidal rule.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Integration_num_trapezes_notation.svg/640px-Integration_num_trapezes_notation.svg.png)