# Riemann sum error and the integral

It is a well known, that we have the following approximation error: $$\left|\int_{a}^{b}f(t)dt-\sum_{i=0}^{n}f\left(\xi_{i}\right)s_{n}\right|<\frac{b-a}{2}s_{n}\cdot\text{max}_{x\in\left[a,b\right]}\left|f'\left(x\right)\right|,$$ where $s_{n}$ is the length of the equidistant decomposition of the interval $\left[a,b\right]$ and $f\in{C^{1}}\left(\left[a,b\right]\right)$. My quesstions are: 1.) How this error estimate can be improved, if $f$ and $f'$ are both Lipschitz continuous? 2.) How such estimates look like, if $f$ is a bivariate function?

Best regards Lucas

• Can anyone tell me any sources (books) where this topic is covered? – Lucas Rimberg Aug 21 '13 at 18:28
• Is $s_n$ about (or exactly) $(b-a)/n$? – marty cohen Jun 1 '15 at 22:36

I guess $(\xi_i)$ is the equidistant decomposition of the interval. Here are hints: 1) consider the case when $f'$ is constant; 2) the estimate is a consequence of the mean value theorem, try to adapt the proof.