Limit of Riemann Sums Problem: In 1, you evaluated x2 dx by using midpoint sums with n =

Chapter 5, Problem 13

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Limit of Riemann Sums Problem: In 1, you evaluated x2 dx by using midpoint sums with n = 6. In this problem you will explore what happens to approximate values of this integral as n gets larger. a. Use your programs to show that L10 = 18.795, U10 = 23.295, M10 = 20.9775, and T10 = 21.045. b. Calculate L100 and L500. What limit does Ln seem to be approaching as n increases? c. Calculate U100 and U500. Does Un seem to be approaching the same limit as Ln? What words describe a function f(x) on the interval [1, 4] if Ln and Un have the same limit as n approaches infinity (and thus as x approaches zero)? d. Explain why the trapezoidal sums are always slightly greater than your conjectured value for the exact integral and why the midpoint sums are always less than your conjectured value.