   Chapter 4.1, Problem 23E

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 ) = sin x ,    0 ≤ x ≤ π

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=sinx,   0xπ

3) Calculation:

Here a=0, b=π

x=b-an

=π-0n

x=πn

Now find out  xi=a+ix

xi=0+i·πn=πin

Therefo

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