Chapter B, Problem 10E

Comparing Riemann Sums Consider a trapezoid of area 4 square units bounded by the graphs of y = x, y = 0, x = 1, and x = 3.

(a) Sketch the graph of the region.

(b) Divide the interval [1, 3] into n equal subintervals and show that the endpoints of the subintervals are

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

(c) Show that the left Riemann sum is

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

(d) Show that the right Riemann sum is

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

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

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

(f) Show that $\underset{n\to \infty }{\lim }{S}_L=\underset{n\to \infty }{\lim }{S}_R=4.$