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18.01 Section, December 8, 2015 Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B) . Riemann sums 1. Estimate R1 t f (t) 0 f (t) dt, where all we know about f is the following: 0 0.8 0.2 1.1 0.4 1.2 0.6 1.0 0.8 0.7 1 0.6 Do this: (a) using rectangles where the left endpoint is the function value; (b) using rectangles where the right endpoint is the function value; (c) using trapezoids. 1 2. In this problem, you will estimate ln(2) based on the fact that R 1 x = ln x. (a) What integral do you need to estimate, in order to estimate ln(2)? (b) Come up with a lower bound for ln(2) by using four bars. Draw a picture. (c) Come up with a upper bound for ln(2) by using four bars. Draw a picture. 3. Suppose Rf ≥ 0 is a function that is zero everywhere except at a finite number of points. ∞ What is −∞ f (x) dx? Justify your answer using Riemann sums. 4. Express Rπ 0 x sin(x) dx as a limit of sums. 2