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Find a definite integral expression that represents the area of the given region in polar plane and then find the exact value of the expression

The region inside both of the cardioidsr=3+3sinθandr=1+sinθ

Short Answer

Expert verified

The area can be given as 0π6(3-3sinθ)2-(1+sinθ)2dθ

And the value of the integral is 1.879.

Step by step solution

01

Given inofrmation

We are given two cardioidsr=3-3sinθandr=1+sinθ

02

Find the area

The area can be given as

0π6(3-3sinθ)2-(1+sinθ)2dθ

On evaluating we get,

0π6(9-18sinθ+9sin2θ-1-2sinθ+sin2θ)dθ=0π6(8-20sinθ+10sin2θ)dθ=[8θ+20cosθ+θ2+sin2θ4]π60=21.879-20=1.879

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