Consider the region bounded by the graphs of y = axⁿ, y = abⁿ, and x = 0, as shown in the figure.

(a) Find the ratio R₁(n) of the area of the region to the area of the circumscribed rectangle.(b) Find lim _{n ∞} R₁(n) and compare the result with the area of the circumscribed rectangle.(c) Find the volume of the solid of revolution formed by revolving the region about the y-axis. Find the ratio R₂(n) of this volume to the volume of the circumscribed right circular cylinder.(d) Find lim _{n ∞} R₂(n) and compare the result with the volume of the circumscribed cylinder.(e) Use the results of parts (b) and (d) to make a conjecture about the shape of the graph of y = axⁿ, 0 ≤ x ≤ b, as n ∞.

Transcribed Image Text:

y ab" {y=ax" b -x