   Chapter 15.3, Problem 17E

Chapter
Section
Textbook Problem

Use a double integral to find the area of the region.17. The region inside the circle (x − 1)2 + y2 = 1 and outside the circle x2 + y2 = 1

To determine

To find: The area of the region using double integral.

Explanation

Given:

The region D lies inside the circle (x1)2+y2=1 and outside x2+y2=1 .

Formula used:

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π , then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

Calculation:

From the given region D, obtain the value of r.

(x1)2+y2=1x2+y2=2xr2=2rcosθr=2cosθ

It is observed that the value of r varies from 1 to 2cosθ and to find limit of θ , substitute 2cosθ=1 , that is θ=cos1(12) .

Therefore, θ varies from π3 to π3 .

Substitute x=rcosθ and y=rsinθ in the equation (1),

DdA=π3π312cosθrdrdθ

Integrate with respect to r and apply the limit as shown below

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