Simpson's Rule on Cubics

Simpson's rule approximates a definite integral

by replacing the integrand f with the quadratic  function that agrees with f at the endpoints and midpoint of each sub-interval.  (For comparison, the Left- and Right-Hand Riemann sums each replace f with a constant function, the Trapezoid and Midpoint rules replace f with the linear function respectively agreeing with f at the endpoints of the interval or agreeing with both f and f' at the midpoint of the interval.)

It is remarkable that Simpson's rule gives the exact values of definite integrals not only for any quadratic but also for any cubic polynomial, using only one sub-interval.

This can be algebraically verified by using the change of variable x = a + (b - a)t and verifying that Simpson's rule with one sub- interval gives the exact value for

Here is a more geometric argument.

Let f be a cubic polynomial, and let q be the quadratic function satisfying f(a) = q(a), f(b) = q(b), and f((a+b)/2) = q((a+b)/2).

Then the error in using Simpson's rule for approximating

is

where E is the cubic polynomial defined by E(x) = q(x) - f(x).

Because  E(a) = E(b) = E((a+b)/2) = 0, the inflection point in the graph of E occurs at x = (a+b)/2.  Cubic polynomials are symmetric about their inflections points, so the region lying between the curve and the x-axis on one side of the inflection point is congruent to the region between the curve and the x-axis on the others side of the inflection point. 

Hence

That is, the approximation has no error.