Show that S₂ also gives the exact value for ∫^b_a x³ dx and conclude, as in Exercise 63, that S_N is exact for all cubic polynomials. Show by counterexample that S₂ is not exact for integrals of x⁴.

Data From Exercise 63

For N even, divide [a, b] into N subintervals of width Δx = b − a/N. Set x_j = a + j Δx, y_j = ƒ(xj), and

Data From Exercise 62

s2= b-a 3N (V2j + 4y2j+1 + y2j+2)