Found an interesting simple approximation for the integral of a function

[ttw]

The geometry seems to fit a trapezoid so I'd guess the rule is quadratic. I did plug in various values for general polynomials (X^k is sufficient) and used (the interval 0 to 1 is sufficient) and tried various values of k. If I did it correctly (not guaranteed, of course), X^2 would be integrated correctly but not X^3. The midpoint rule and composite trapezoidal rules do the same for fewer function evaluations.