   Chapter 4.1, Problem 21E

Chapter
Section
Textbook Problem

# Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f ( x ) = 2 x x 2 + 1 ,    1 ≤ x ≤ 3

To determine

To find:

An expression for the area under the graph.

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, so the width of each n strip is

x=b-an

where x0=a  and  xn=b . The right endpoints of the subintervals are xi=a+ix

2) Given:

fx=2xx2+1,   1x3

3) Calculation:

Here a=1, b=3

x=b-an

=3-1n

x=2n

Now find out  xi=a+ix

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