   Chapter 10.5, Problem 17E

Chapter
Section
Textbook Problem

# Finding the Area of a Polar Region In Exercises 19-26, use a graphing utility to graph the polar equation. Find the area of the given region analytically.Inner loop of   r = 1 + 2 cos θ

To determine

To Calculate: The graph of polar equation r=1+2cosθ by graphing utility

And also, area of inner loop of r=1+2cosθ

Explanation

Given:

The provided polar equation r=1+2cosθ

Calculation:

Graph of the curve r=1+2cosθ is shown below:

To find the α and β take r=0

That is,

0=1+2cosθ

cosθ=12

θ=2π3,4π3

So, α=2π3 and β=4π3.

Therefore, the total area will be:

A=2.122π34π3[r]2.dθ

A=2π34π3[1+2cosθ]2.dθ

A=2π34π3[1+4cos2θ+4cosθ]

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