# bound on error of Simpson’s rule

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

 $\left|\int\limits_{a}^{b}f(x)\,dx-\!\left(\!\frac{b-a}{3n}\!\right)\!\left(\!f% (a)\!+\!f(b)\!+\!4\sum_{j=1}^{\frac{n}{2}}f\!\left(\!a\!+\!\frac{(b-a)(2j-1)}{% n}\!\right)\!+\!6\sum_{j=1}^{\frac{n-2}{2}}f\!\left(\!a\!+\!\frac{(b-a)(2j)}{n% }\!\right)\!\right)\!\right|\leq\frac{M\!(b-a)^{5}}{180n^{4}}.$
Title bound on error of Simpson’s rule BoundOnErrorOfSimpsonsRule 2013-03-22 15:57:52 2013-03-22 15:57:52 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Result msc 41A55 msc 28-00