bound on error of Simpson’s rule

If $f$ is Riemann integrable on $[a,b]$ and $|f^{(4)}(x)|\le M$ for all $x\in [a,b]$, then

$$\left|\underset{a}{\overset{b}{\int }}f(x)𝑑x-\left(\frac{b-a}{3n}\right)\left(f(a)+f(b)+4\sum _{j=1}^{\frac{n}{2}}f\left(a+\frac{(b-a)(2j-1)}{n}\right)+6\sum _{j=1}^{\frac{n-2}{2}}f\left(a+\frac{(b-a)(2j)}{n}\right)\right)\right|\le \frac{M{(b-a)}^5}{180n^4}.$$

Title	bound on error of Simpson’s rule
Canonical name	BoundOnErrorOfSimpsonsRule
Date of creation	2013-03-22 15:57:52
Last modified on	2013-03-22 15:57:52
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	7
Author	Wkbj79 (1863)
Entry type	Result
Classification	msc 41A55
Classification	msc 28-00