f is a continuous real-valued function on the closed interval [0, 1] such that f(1) = 0. A point (a1, a2, … , an) is chosen at random from the n-dimensional region 0 < x1 < x2 < … < xn < 1. Define a0 = 0, an+1 = 1. Show that the expected value of ∑0n (ai+1 – ai) f(ai+1) is ∫01 f(x) p(x) dx, where p(x) is a polynomial of degree n which maps the interval [0, 1] into itself (and is independent of f).