   Chapter 15.3, Problem 15E

Chapter
Section
Textbook Problem

Use a double integral to find the area of the region.15. One loop of the rose r = cos 3θ

To determine

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

Explanation

Given:

The region D is r=cos3θ .

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 r varies from 0 to cos3θ and to find the limits of θ , substitute cos3θ=0 , that gives 3θ=π2,π2 .

Therefore θ varies from π6 to π6 .

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

DdA=π6π60cos3θrdrdθ

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

π6π60cos3θrdrdθ=π6π6[r22]0cos3θd

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