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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

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).

xcos2xsinxf(x)
01.000.000.00
π60.750.500.38
π30.250.870.22
π20.001.000.00
2π30.250.870.22
5π60.750.500.38
π1.000.000.00
7π60.75-0.50-0.38
4π30.25-0.87-0.22
3π20.00-1.000.00
5π30

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