   Chapter 15.3, Problem 18E

Chapter
Section
Textbook Problem

Use a double integral to find the area of the region.18. The region inside the cardioid r = 1 + cos θ and outside the circle r = 3 cos θ

To determine

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

Explanation

Given:

The region D lies inside the cardioid r=1+cosθ and outside the circle r=3cosθ .

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, it is observed that one part of the region D lies in the first quadrant say A1 and another part lies in second quadrant say A2 .

Now for A1 , the value of r varies from 3cosθ to 1+cosθ and to find limit of θ , substitute 3cosθ=1+cosθ , that is θ=cos1(12) .

Therefore, θ varies from π3 to π2 .

Similarly, for A2 , the value of r varies from 0 to 1+cosθ and θ varies from π2 to π and multiply both by 2. Therefore, by (1) the area of given region becomes,

DdA=A1+A2=π3π23cosθ1+cosθrdrdθ+π2π01+cosθrdrdθ

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

π3π23cosθ1+cosθrdrdθ+π2π01+cosθrdrdθ=π3π2[r22]3cosθ1+cosθdθ+π2π[r22]01+cosθdθ=π3π2[(1+cosθ)22(3cosθ)22]dθ+π2π[(1+cosθ)22(0)22]dθ=[π3π2(12+2cosθ2+cos2θ29cos2θ2)dθ+π2π[12+2cosθ2+cos2θ2]dθ]=π3π2(12+cosθ4cos2θ)dθ+π2π[12+cosθ+cos2θ2]dθ

Simplify the terms as follows

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