Chapter 8.3, Problem 71E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Area In Exercises 71 and 72, find the area of the given region. y = sin x , y = sin 3 x

To determine

To calculate: The area of given region of the functions y=sinx , y=sin3x.

Explanation

Given: y=sinxÂ ,Â y=sin3x

Refer the image in the giving question.

Calculation:

Consider the area in the given region,

y=sinxÂ ,Â y=sin3x

Refer the image in the giving question.

From the graph, the limit of x = 0,Ï€2. You can also find the limit of x by algebraic calculation.

Now letâ€™s find the intersection point by equating y.

sinx=sin3xsinxâˆ’sin3x=0sinx(1âˆ’sin2x)=0

Simplify further,

sinx=0 or sin2x=1.

x=0 or x=Ï€2

So that the limit of x are,

x=0Â ,Â Ï€2

Hence the area of region is,

A=âˆ«0Ï€2(sinxâˆ’sin3x)dx

Use the trigonometry identity,

sin3x=3sinxâˆ’4sin3xsin3x=14(3sinxâˆ’sin3x)

So,

A=âˆ«0Ï€2

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