Chapter B, Problem 9E

Comparing Riemann Sums Consider a triangle of area 2 square units bounded by the graphs of y = x, y = 0, and x = 2.

(a) Sketch the graph of the region.

(b) Divide the interval [0, 2] into n equal subintervals and show that the endpoints of the subintervals are

$0<1\left(\frac{2}{n}\right)<...<\left(n-1\right)\left(\frac{2}{n}\right)<n\left(\frac{2}{n}\right).$

(c) Show that the left Riemann sum is

$S_L={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left[\left(i-1\right)\left(\frac{2}{n}\right)\right]}\left(\frac{2}{n}\right).$

(d) Show that the right Riemann sum is

$S_R={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left[i\left(\frac{2}{n}\right)\right]}\left(\frac{2}{n}\right).$

(e) Use a graphing utility and the program in Example 4 to complete the table.

n	5	10	50	100
Left sum, SL
Left sum, SR

(f) Show that $\underset{n\to \infty }{\lim }{S}_L=\underset{n\to \infty }{\lim }{S}_R=2.$