   Chapter 5.1, Problem 24E

Chapter
Section
Textbook Problem

# Sketch the region enclosed by the given curves and find its area. y = cos x ,   y = 1 − cos x ,   0 ≤ x ≤ π

To determine

To:

Sketch the region and find the enclosed area

Explanation

1) Concept:

Formula:

The area A of the region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b  is

A= abfx-gxdx

fx-gx=fx-gx when fxg(x)gx-fx when gxf(x)

2) Given:

y=cosx and   y=1-cosx,  0xπ

3) Calculation:

The point of intersection occurs when both the equations are equal to each other, that is,

cosx=1-cosx

2cosx=1

cosx=12

x=π3 when   0xπ

Thus from the graph the points of intersection is at x=0,  x=π3 and   x=π. The region is sketched in the following figure.

Here, A=A1+A2

Where A1 is the area between the curves from  0xπ3

And A2 is the area between the curves from π3xπ

Here,cosx1-cosx when   0xπ3. Therefore,

fx=cosx

gx=1-cosx

Therefore, the required area is

A1=0π3cosx-1-cosxdx

A1=0π3cosx-1+cosx dx

A1=0π32cosx-1 dx

Compute the integral using the standard integration rule

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