   Chapter 5, Problem 44RE ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Graph the function f(x) = cos2x sin x and use the graph to guess the value of the integral ∫ 0 2 π f ( x )   d x . Then evaluate the integral to confirm your guess.

To determine

To Draw: the graph for the function f(x)=cos2xsinx.

To Evaluate: The value of the integral function 02πf(x)dx

Explanation

Given information:

The function is f(x)=cos2xsinx.

The region lies between x=0 and x=2π.

Calculation:

Show the function as follows:

f(x)=cos2xsinx (1)

Substitute 0 for x in Equation (1).

f(0)=cos2(0)sin(0)=1×0=0

Substitute π6 for x in Equation (1).

f(π6)=cos2(π6)sin(π6)=0.75×0.5=0.38

Calculate the remaining values of the function as shown below.

Show the x and f(x) values as in Table (1).

 x cos2x sinx f(x) 0 1.00 0.00 0.00 π6 0.75 0.50 0.38 π3 0.25 0.87 0.22 π2 0.00 1.00 0.00 2π3 0.25 0.87 0.22 5π6 0.75 0.50 0.38 π 1.00 0.00 0.00 7π6 0.75 -0.50 -0.38 4π3 0.25 -0.87 -0.22 3π2 0.00 -1.00 0.00 5π3 0

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