   Chapter 4.1, Problem 22E

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 ) = x 2 + 1 + 2 x ,   4 ≤ x ≤ 7

To determine

To find:

An expression for 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 end points of the subintervals are xi=a+ix.

2) Given:

fx=x2+1+2x,   4x7

3) Calculation:

Here a=4, b=7

x=b-an

=7-4n

x=3n

Now find out  xi=a+ix

xi=4+i·3n=4+3in

Therefore, the area under the function f(x) is given by

A=limni=1nfxi

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