   Chapter 4.1, Problem 27E

Chapter
Section
Textbook Problem

# Let A be the area under the graph of an increasing continuous function f from a to h, and let Ln and Rn be the approximations to A with n subintervals using left and right endpoints, respectively.(a) How are A, Ln, and Rn related?(b) Show that R n − L n = b − a n [ f ( b ) − f ( a ) ] Then draw a diagram to illustrate this equation by showing that the n rectangles representing Rn - Ln can be reassembled to form a single rectangle whose area is the right side of the equation.(c) Deduce that R n − A < b − a n [ f ( b ) − f ( a ) ]

To determine

a)

To describe:

A relation

Explanation

1) Concept:

The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles.

A=limnRn=limnfx1x+fx2x++fxnx=limni=1nfxix

The width of the interval a, b is (b-a)

To determine

b)

To show:

Rn-Ln=b-anfb-fa

To determine

c)

To deduce:

Rn-A<b-anfb-fa

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