   Chapter 4.2, Problem 75E

Chapter
Section
Textbook Problem

# Find ∫ 1 2 x − 2 d x . Hint: Choose x i * to be the geometric mean of x i − 1 and x i (that is, x i * = x i − 1 x i ) and use the identity 1 m ( m + 1 ) = 1 m − 1 m + 1

To determine

To find:

The 12x-2dx by choosing xi* to be the geometric mean of xi-1 and xi and use the identity 1mm+1=1m-1m+1

Explanation

1) Concept:

Use theorem (4) to find the definite integral.

Theorem (4):

If f is integrable on [a, b], then

abfxdx=limni=1nfxi* x

Where

x= b - an and xi* is in [xi-1,xi] where xi=a+i x

2) Formula:

i=1n(ai-bi)= i=1nai-i=1nbi

3) Given:

12x-2dx

And

1mm+1=1m-1m+1

4) Calculation:

Here, a=1 and b=2

x=b-an=2-1n=1n

Thus,xi=a+i x =1+ in=1+in

And

xi-1=1+i-1n

Choose xi* to be the geometric mean of xi-1 and xi that is

xi*=xi-1xi Thus

xi*=xi-1xi= 1+i-1n1+in

By using theorem (4)

12x-2dx=limni=1nfxi* x

=limn1ni=1n1xi*2

Substitute,

xi*= 1+

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