Chapter 4.1, Problem 27E

Let A be the area under the graph of an increasing continuous function f from a to b, and let L_n and R_n be the approximations to A with n subintervals using left and right endpoints, respectively.

(a) How are A, L_n, and R_n related?

(b) Show that

$R_n-L_n=\frac{b-a}{n}\left[f\left(b\right)-f\left(a\right)\right]$

Then draw a diagram to illustrate this equation by showing that the n rectangles representing R_n – L_n can be reassembled to form a single rectangle whose area is the right side of the equation.

(c) Deduce that

$R_n-A<\frac{b-a}{n}\left[f\left(b\right)-f\left(a\right)\right]$