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Quiz 3 for MATH 105 SECTION 205 February 04, 2015 Given Name Family Name Student Number Z 2 1. (a) (1 point) Write down the midpoint Riemann sum with n = 50 to approximate 1 notations. 1 dx using sigma x (a) (b) (0.5 points) Express 2 + 4 + 6 + 8 + · · · + 80 using sigma notations. (b) Z (c) (1 point) Let f and g be two integrable functions on [1, 2], if Z 2 compute [2f (x) − 3g(x)] dx. 2 Z 2 g(x) dx = 1, f (x) dx = 2 and 1 1 1 (c) √ 1 − x2 , if 0 ≤ x ≤ 1, 2. Let f (x) = , and R be the region bounded by the graph of f (x) and x-axis between 2 − 2x, if x > 1, x = 0 and x = 3, then (a) (1 point) Find the area of R. (a) (b) (1 point) Find the net area of R. (b) Z (c) (0.5 points) Compute 3 f (x) dx. 0 (c) 3. Let f (x) = 1, if x is a rational number , then 0, if x is not a rational number (a) (1. points) What’s the value for the left Riemann sum for any regular partition of [0, 1]?. (a) (b) (0.5 points) For any partition of [0, 1], is it true that we can always find a Riemann sum whose value is 0? (Just put ‘Yes’ or ‘No’). (b) (c) (0.5 points) Is f integrable on [0, 1]? (Just put ‘Yes’ or ‘No’). (c) Z 4. (3 points) Use the Riemann sum to compute 1 (2x + 1) dx. 0 Your Score: /10