Editing Simpson's rule (section)

==== Averaging the midpoint and the trapezoidal rules ====
Another derivation constructs Simpson's rule from two simpler approximations: the [[rectangle rule|midpoint rule]]
<math display="block">M = (b - a)f\left(\frac{a + b}{2}\right)</math>
and the [[trapezoidal rule]]
<math display="block">T = \frac{1}{2} (b - a)\big(f(a) + f(b)\big).</math>

The errors in these approximations are
<math display="block">\frac{1}{24} (b - a)^3 f''(a) + O\big((b - a)^4\big)</math>
and
<math display="block">-\frac{1}{12} (b - a)^3 f''(a) + O\big((b - a)^4\big)</math>
respectively, where <math>O\big((b - a)^4\big)</math> denotes a term asymptotically proportional to <math>(b - a)^4</math>. The two <math>O\big((b - a)^4\big)</math> terms are not equal; see [[Big O notation]] for more details. It follows from the above formulas for the errors of the midpoint and trapezoidal rule that the leading error term vanishes if we take the [[weighted average]]
<math>\frac{2M + T}{3}.</math>
This weighted average is exactly Simpson's rule.

Using another approximation (for example, the trapezoidal rule with twice as many points), it is possible to take a suitable weighted average and eliminate another error term. This is [[Romberg's method]].