Let p(x) be a cubic polynomial, and q(x) a quadratic polynomial.
Part A: Show that if p(-1)=q(-1) and p(1)=q(1) then the integral ∫I p(x)-q(x)dx = (4/3)(p(0)-q(0)), where I is the interval [-1,1].
Part B: Show that for an arbitrary interval I=[α,β], if p(α)=q(α) and p(β)=q(β) then the integral ∫I p(x)-q(x)dx = 2(β-α)(p(γ)-q(γ))/3, where γ=(α+β)/2.