Consider the region bounded by the graphs of f (x) = 8x/(x + 1), x =0, x = 4, and y = 0, as shown in the figure. To print an enlarged copy of the graph, go to MathGraphs.com.(a) Redraw the figure, and complete and shade the rectangles representing the lower sum when n = 4. Find this lower sum.(b) Redraw the figure, and complete and shade the rectangles representing the upper sum when n = 4. Find this upper sum.(c) Redraw the figure, and complete and shade the rectangles whose heights are determined by the function values at the midpoint of each subinterval when n = 4. Find this sum using the Midpoint Rule.(d) Verify the following formulas for approximating the area of the region using n subintervals of equal width.

(e) Use a graphing utility to create a table of values of s(n), S(n), and M(n) for n = 4, 8, 20, 100, and 200.(f) Explain why s(n) increases and S(n) decreases for increasing values of n, as shown in the table in part (e).

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