   Chapter 7.8, Problem 47E

Chapter
Section
Textbook Problem

# (a) If g ( x ) = ( sin 2 x ) / x 2 , use your calculator or computer to make a table of approximate values of ∫ 1 t g ( x )   d x for t = 2 , 5 , 10 , 100 , 1000 ,  and 10, 000 . Does it appear that ∫ 1 ∞ g ( x )   d x is convergent?(b) Use the Comparison Theorem with f ( x ) = 1 / x 2 to show that ∫ 1 ∞ g ( x )   d x is convergent.(c) Illustrate part (b) by graphing f and g on the same screen for 1 ≤ x ≤ 10 . Use your graph to explain intuitively why ∫ 1 ∞ g ( x )   d x is convergent.

To determine

( a);

To make:

A table of approximate values of 1tg(x)dx using calculator or computer and state if it appear convergent.

Explanation

Given:

g(x)=sin2xx2

Consider g(x)=sin2xx2

Now, put t=2,5,10,100,1000,10000 and get the table

To determine

(b)

To show:

that the integral 1g(x)dx is convergent using comparison theorem.

To determine

(c)

To illustrate:

Part (b) by graphing  f and g on the same screen for 1x10 and explain intuitively why 1g(x)dx is convergent.

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