Evaluate the limit by first interpreting the sum as a Riemann sum for a function defined on [0, 9]. DO NOT APPROXIMATE

Question

lim n→∞ (9/n) (sqrt(9/n) + sqrt(18/n) + sqrt(27/n) +....+sqrt(9n/n))

Answer 1

This is limit of right Reaman sum and it equel ∫₀⁹√xdx = 2/3x^3/2₀⁹ = 18

Answer 2

This is a right-hand-side Riemann sum from 0 to 9 with an interval of (9-0)/n = 9/n. 9/n is the width of the rectangles and the height for rectangle i will be f(i*9/n). Therefore, as n goes to infinity, this will approach the integral from 0 to 9 of f(x) = sqrt(x).

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