Chapter 10.5, Problem 20E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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 = 2 − 4 cos θ

To determine

To Calculate: The graph of polar equation r=24cosθ by the use of graphing utility

And also, area of inner loop of r=24cosθ.

Explanation

Given:

The polar equation r=2âˆ’4cosÎ¸.

Calculation:

The graph of the curve r=2âˆ’4cosÎ¸ is:

To find the value of Î± and Î² take r=0.

That is,

â‡’0=2âˆ’4cosÎ¸

â‡’cosÎ¸=12

Î¸=Ï€3

So, Î±=0 and Î²=Ï€3

Therefore, the total area will be:

A=2.12âˆ«0Ï€3[r]2.dÎ¸

A=âˆ«0Ï€3[2âˆ’4cosÎ¸]2.dÎ¸

A=âˆ«0Ï€3[4+16cos2Î¸âˆ’16cosÎ¸].dÎ¸

A=âˆ«0Ï€3[4+16(1+cos2Î¸2)âˆ’16cosÎ¸]

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