Rectangles, trapezoids, and parabolas make different guesses about what happens between samples.
Composite Simpson’s rule uses an even number of subintervals and fits quadratics across pairs. It is exact for polynomials through degree three in exact arithmetic. Here f(x) = x³ − x; the exact integral comes from x⁴/4 − x²/2.
More samples do not turn every numerical estimate into an exact answer.
Simpson’s rule uses quadratic interpolants across pairs and is exact for this cubic, up to roundoff.