   Chapter 5.1, Problem 31E

Chapter
Section
Textbook Problem

# Sketch the region enclosed by the given curves and find its area. y = cos 2 x   sin x ,   y = sin x ,   0 ≤ x ≤ π

To determine

To:

The sketch the region enclosed by the given curves and find its 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=cos2xsinx,  y=sinx,  0xπ

3) Calculation:

First, find the intersection points of the curves by solving their equations simultaneously.

cos2xsinx=sinx

This gives,

cos2xsinx-sinx=0

sinxcos2 x-1=0

sinxsin2x=0

Therefore,

sinx=0

Which gives the boundary points of the region as

x=0  and π

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