   Chapter 15.3, Problem 16E

Chapter
Section
Textbook Problem

Use a double integral to find the area of the region.16. The region enclosed by both of the cardioids r = 1 + cos θ and r = 1 − cos θ

To determine

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

Explanation

Given:

The region D is enclosed by both of the cardioids r=1+cosθ and r=1cosθ .

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 it is enough to find the area in the first quadrant only and multiply it with 4.

Here, the value of r varies from 0 to 1cosθ and the value of θ varies from 0 to π2 .

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

DdA=0π201cosθrdrdθ

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

0π201cosθrdrdθ=0π2[r22]01cosθdθ=0π2[(1cosθ)22022]dθ=12

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