APPM 1345 Midterm: appm1345spring2016exam2_sol
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Instructions: books, notes, and electronic devices are not permitted. 2: (16 points) using the de nition of a de nite integral, i. e. the limit of a riemann sum, If needed, recall: n n(n + 1) i = 2 (cid:21)2 n (cid:19) +(cid:18) n + 2i n2 + 4in + 4i2 n (cid:19)2# (cid:21) i2(cid:21) 4 n i + 1 + i + lim n . 8 n3 (cid:21) n(n + 1)(2n + 1) = lim n (cid:20) 3n n (cid:20)3 + 5 + A (b) z a (c) g(x) =z 3x. 2z 1 u = cos x = du = sin xdx. Z 1 (b) z a f (x) = x a2 x2 is an odd function, therefore the rst integral is 0. The second integral is a semi-circle of radius a, therefore a = 1 xpa2 x2dx + 3z a.
