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Trapezoid Rule

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

The approximation

$\displaystyle \int_a^b f(x)\, dx \approx s_h f = h (f(a)/2 + f(a+h) + \cdots +
f(b-h) + f(b)/2)

refers to the integral as a sum of trapezoids.


For a twice continuously differentiable function, the error can be estimated via

$\displaystyle s_h f - \int_a^b f =
\frac{b-a}{12} f\,'\,'(r) h^2,

with $ r\in[a,b]$ .

More precisely, the error for smooth functions bears the asymptotic expansion

$\displaystyle s_h f - \int_a^b f = c_1(f^\prime(b)-f^\prime(a))h^2 + c_2(f^{\prime\prime\prime}(b)-f^{\prime\prime\prime}(a)) h^4 + \dots

with constants $ c_j$ independent from $ f$ and $ h$ . Thus the trapezoid rule is very exact for $ (b-a)$ -periodic functions. The error strives faster towards zero than any $ h$ -potence.
(Authors: Höllig/Hörner/Abele)


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  automatically generated 4/ 7/2008