University of Illinois at Urbana Champaign Fall

University of Illinois at Urbana-Champaign Fall 2006 Math 444 Group E13 Integration : correction of the exercises. 1. (a) Assume that f : [a, b] ? R is a continuous function such that f(x) ≥ 0 for all x ? (a, b), and ∫ b a f(t)dt = 0. Show that f(x) = 0 for all x ? [a, b] ; can you use the fundamental theorem of calculus to prove this result ? (b) Use this to show that if f is continuous on [a, b] and ∫ b a f(t)dt = 0 then there must exist t ? (a, b) such that f(t) = 0. Correction. (a) First,notice that, since f is continuous, proving that f(t) = 0 for all t ? [a, b] is the same as proving that f(t) = 0 for all t ? (a, b). Now, let us prove the contraposite of the result we are interested in ; in other words, let us prove that if f(x) > 0 for some x ? (a, b), f(x) ≥ 0 for all x ? (a, b) and f is continuous on [a, b] then ∫ b a f(t)dt > 0.

• induction using


• x? ?


• additivity theorem


• riemann sum


• then pick


• gives limc?a ∫


• since ti


• there exists