What is the order of the midpoint rule?

Question

If the Trapezoidal-Rule has the order $n=1$, and Simpson's has order $n=2$, what is the order $n$ of the midpoint rule?

And if the weights of the Trapeziumrule are ($1/2, 1/2$) and those of the Simpson-Rule are ($1/6, 4/6, 1/6$), what are the weights of the midpoint-rule?

Thank you in advance!

Answer 1

The error of the Midpoint rule is

$Error_m \leq \frac{k(b-a)^3}{24n^2 } = k(b-a)h^2/24$. So it equals to $O(h^2)$, where $k = max|f"|$.

Answer 2

The order of the midpoint rule is $n=2$. For examples, you can see Why does Trapezoidal Rule have potential error greater than Midpoint?