   Chapter 8.5, Problem 54E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Finding the Area of a Region In Exercises 53-56, sketch the region bounded by the graphs of the functions and find the area of the region. y = sin   x ,   y   = cos   2 x ,   x = − π 2 ,   x = π 6

To determine

To graph: The region bounded by the functions, y=sinx and y=cos2x for x=π2 to x=π6 and also calculate the area of the region.

Explanation

Given Information:

The provided functions are y=sinx and y=cos2x for x=π2 to x=π6.

Formula used:

Write the formula of definite integral of abcosnudu.

abcosnudu=[sinnun]ab

Here, n is the any positive integer.

Write the formula of definite integral of absinudu.

absinudu=[cosu]ab

Graph:

Draw area bounded by graph of y=sinx and y=cos2x for x=π2 to x=π6.

Consider provided function,

y=sinx

At x=π2,

yx=π2=sin(π2)=sin(π2)=1

At x=π6,

yx=π6=sin(π6)=sin(π6)=0.50

At x=0,

yx=0=sin(0)=0

At x=π6,

yx=π6=sin(π6)=sin(π6)=0.50

Now, the table for ordered pair (x,y) for the function y=sinx is shown below:

xyπ21π60.5000π60.50

Again, consider provided function.

y=cos2x

At x=π2,

yx=π2=cos2(π2)=cos(π)=1

At x=π6,

yx=π6=cos2(π6)=cos(π3)=0.5

At x=0,

yx=0=cos2(0)=cos0=1

At x=π6,

yx=π6=cos2(π6)=cos(π3)=0.50

Now, the table for ordered pair (x,y) for the function y=cos2x is shown below:

xyπ21π60

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