   Chapter 10, Problem 108RE

Chapter
Section
Textbook Problem

Finding the Area of a Polar Region In Exercises 103-108, use a graphing utility to graph the polar equation. Find the area of the given region analytically.Common interior of r = 4 cos θ and r = 2

To determine

To calculate: The area of the region analytically. Also graph the common interior of the polar equation r=4cosθ and r=2 using a graphing utility.

Explanation

Given:

The polar equations are r=4cosθ and r=2.

Calculation:

Consider the equations, r=4cosθ and r=2.

Draw the graph of the equations, r=4cosθ and r=2 using Ti83 calculator. The following steps are needed.

Step 1. Press on button.

Step 2. Press MODE button and choose ‘funct” and in funct choose Pol.

Step 3. Press Y= button and write r1=4cosθ and r2=2

Step 4. Press GRAPH button.

And the graph of the polar equation r=4cosθ and r=2 is obtained as below:

The area of the region between the circle r=2 between θ=π3 and θ=π2 is

A1=12π3π2(2)2dθ

So,

A1=12[2π3π2dθ ]

That is,

A1=[θ]π3π2

Apply the limits with the help of the property:

[F(x)]ab=F(b)F(a)

Then,

A1=π2π3=2π6

The area of the region between the circle r=4cosθ and r=2 can be obtained by integrating from θ=0 and θ=π3 is:

A2=120π3(4cosθ)2dθ

So,

A2=120π3<

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