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In Exercises 29–38, find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral.

The region bounded by the limaconr=1+ksinθ, where0<k<1. Explain why it makes sense for the area to approach πask0.

Short Answer

Expert verified

The iterated integral that represents the area of the given region isπ+πk22

Step by step solution

01

Given Information

Given equation : r=1+ksinθ

02

Graphing the strophoid and find the area bounded by the loop of the graph

First, we plot the curve r=1+ksinθwithk=12:

03

Finding an iterated integral that represents the area of the given region

The arc can be represented as

A=2-π/2π/212r2dθA=-π/2π/2(1+ksinθ)2dθ=-π/2π/2(1+2ksinθ+k2sin2θ)dθ=-π/2π/2(1+2ksinθ+k22(1-cos2θ)dθ

Integrate in relation to θ:

A=(1+2ksinθ+k22(1-cos2θ)-π/2π/2

Pointing the limits,

localid="1651493633771" role="math" A=π22kcosπ2+k22π212sin(π)π22kcosπ2+k22π212sin(π)A=π2+k22π2π2+k22π2A=π+πk22

As a result, the limacon's area is A=π+πk22where k0,A=π

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