calculus - 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