Simpson's rule has error scaling like h⁴ against the trapezoidal rule's h². Why does fitting parabolas instead of straight lines gain two whole orders?

D. Because a parabola matches one more derivative, so a term cancels.

A parabola matches one more derivative than a trapezoid, killing the leading error term, and it kills more than expected, since Simpson's is exact for cubics, the odd error term cancelling between the two halves of each panel. Halving h reduces trapezoidal error by 4 and Simpson's by 16. The gain evaporates at kinks, where the extra derivatives do not exist.