For the left and right Riemann sum, an error bound is given by:
|Error| $\leq \frac{(B-A)^2}{2N} M_1$ where $M_1 = \max_{x \in [A,B]}f'(x)$.
I can't seem to find any information on the error bounds for the upper and lower Riemann sums however. Are they the same or different?
Best Answer
Let $t_k$ be the evaluation point on the interval $[x_k,x_{k+1}]$ where $x_{k+1}-x_k=h$, $Nh=B-A$. Then on that interval $$ |f(x)-f(t_k)|\le M_1·|x-t_k| $$
In the integral over that interval this gives \begin{align} \left|\int_{x_k}^{x_{k+1}}f(x)dx-f(t_k)(x_{k+1}-x_k)\right| &\le\int_{x_k}^{x_{k+1}}|f(x)-f(t_k)|dx \\&\le M_1·\int_{x_k}^{x_{k+1}}|x-t_k|dx \\&=M_1·\frac{(x_{k+1}-t_k)^2+(x_k-t_k)^2}2\le M_1·\frac{(x_{k+1}-x_k)^2}2 \end{align} This shows that the claimed bound is valid for Riemann sums of all kinds.