Chapter 4.2, Problem 69E

Graphical Reasoning 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.

Lower sum: $s(n)={\displaystyle \underset{i=1}{\overset{n}{\sum }}f\left[(i-1)\frac{4}{n}\right]\left(\frac{4}{n}\right)}$

Upper sum: $S(n)={\displaystyle \underset{i=1}{\overset{n}{\sum }}f\left[(i)\frac{4}{n}\right]\left(\frac{4}{n}\right)}$

Midpoint Rule: $M(n)={\displaystyle \underset{i=1}{\overset{n}{\sum }}f\left[\left(i-\frac{1}{2}\right)\frac{4}{n}\right]\left(\frac{4}{n}\right)}$

(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).