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Concept
Trapezoidal rule
Summary
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for approximating the definite integral.
The trapezoidal rule works by approximating the region under the graph of the function
as a trapezoid and calculating its area. It follows that
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.
A 2016 Science paper reports that the trapezoid rule was in use in Babylon before 50 BCE for integrating the velocity of Jupiter along the ecliptic.
When the grid spacing is non-uniform, one can use the formula
wherein
For a domain discretized into equally spaced panels, considerable simplification may occur. Let
the approximation to the integral becomes
The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result:
There exists a number ξ between a and b, such that
It follows that if the integrand is concave up (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value.
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