Why does Trapezoidal Rule have potential error greater than Midpoint? I can approximate the area beneath a curve using the Midpoint and Trapezoidal methods, with errors such that:
$Error_m \leq \frac{k(b-a)^3}{24n^2}$ and $Error_T \leq \frac{k(b-a)^3}{12n^2}$.
Doesn't this suggest that the Midpoint Method is twice as accurate as the Trapezoidal Method?

On an interval where a function is concave-down, the Trapezoidal Rule will consistently underestimate the area under the curve.  (And inversely, if the function is concave up, the Trapezoidal Rule will consistently overestimate the area.)

With the Midpoint Rule, each rectangle will sometimes overestimate and sometimes underestimate the function (unless the function has a local minimum/maximum at the midpoint), and so the errors partially cancel out.  (They exactly cancel out if the function is a straight line.)

