




Why these weigths? Because they are exactly correct even for n = 2 if f is a cubic polynomial.
Suppose: f(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3}
Calculate using Simpson's rule with n = 2.
Thus Simpson's rule, with n = 2 is the exact integral on any cubic!_{ }
As a consequence of this fact, if we ide our interval into pieces in each of which f is not far from a cubic, Simpson's rule will be very accurate.