MATA37H3 Lecture Notes - Lecture 4: In C, Antiderivative
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Mata37 - lecture 4 - integrability reformulation and inde nite integrals. Want to show > 0, p of [a, b] s. t. U (f, p ) l(f, p ) < is false, or > 0, p of [a, b] s. t. U (f, p ) l(f, p ) (cid:90) 1. Choose > 0 (chosen later in the process of solving) Let p = {x0 = a < x1 < x2 < < xi 1 < xi < < xn = b} be arbitrary. Mi = inf {g(x)|x [xi 1, xi]} by de nition. Mi = sup{g(x)|x [xi 1, xi]} by de nition. G(x) = either 4 or 0, so mi = 4, mi = 0. U (g, p ) l(g, p ) = Mi(xi xi 1) n(cid:88) mi(xi xi 1) by de nition of u, l n(cid:88) 0(xi xi 1) n(cid:88) n(cid:88) i=1 i=1 (xi xi 1) i=1 n(cid:88) i=1 n(cid:88)
