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Find the area of the region between the loops of the limaconr=1+2cosθ

Short Answer

Expert verified

The region between the limacon's two loops has a surface area isA=π-1

Step by step solution

01

Given information

Consider the limacon isr=1+2cosθ

02

Evaluating the integral

The goal of this issue is to discover an iterated integral in polar coordinates that represents the area of a given location in the polar plane, then evaluate it.

Pointing limicon,r=1+2cosθ

03

Pointing the Graph of integral

Graph ofr=1+2cosθ

The integral of limicon isr=1+2cosθ

For, r=0,1+2cosθ=0cosθ=12

Pole tangents areθ=π4andθ=3π4

Area of small loop isπ4θπ4

Area of large loop is3π4θ3π4

The area of the limacon's region between the two loops is equal to A= Area of Big loop - Area of small loop

A=3π/43π/4r2203+2cosθπ/4π/4r22θ3+2cosθ

04

Calculations 

Integrate for the first time with respect to r

A=3π/43θ/4r2201+2cosθπ/4π/4r2201+2cosθA=3π/43π/4(1+2cosθ)22π/4π/4(1+2cosθ)22θA=3π/43π/41+22cosθ+2cos2θ2θπ/4π/41+22cosθ+2cos2θ2A=3π/43π/4(1+22cosθ+1+cos2θ)2π/4π/4(1+22cosθ+1+cos2θ)2

Integrate in relation to θ,

A=3π/43/4θ+22sinθ+θ+12sin2θ2θπ/4π/4θ+22sinθ+θ+12sin2θ2θA=2θ+22sinθ+12sin2θ23π/43π/42θ+22sinθ+12sin2θ2π/4π/4A=θ+2sinθ+14sin2θ3π/43π/4θ+2sinθ+14sin2θπ/4π/4

Pointing the limits,

A=3π4+2sin3π4+14sin3π23π4+2sin3π4+14sin3π2π4+2sinπ4+14sinπ2π4+2sinπ4+14sinπ2A=3π4+1143π41+14π4+1+14π4114A=π1

As a result, the area of the reaction between the two limacon loops isA=π-1

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