   Chapter 10, Problem 99RE

Chapter
Section
Textbook Problem

# Finding the Area of a Polar Region In Exercises 97-100, find the area of the region.Interior of r = 2 + cos θ

To determine

To calculate: The area of the interior region of the curve r=2+cosθ.

Explanation

Given:

The polar curve is r=2+cosθ.

Calculation:

Consider, r=2+cosθ

First draw the graph.

To graph polar equation, use Ti83 calculator. The steps are as follows

Step 1. Press on button.

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

Step 3. prees Y= button and write r1=2+cosθ

Step 4. Press GRAPH button.

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

Here, the curve of r=2+cosθ is symmetrical about the initial line.

Therefore, the required area is twice the area above the initial line. The limits of integration are θ=0 to θ=π.

The area of interior of a polar curve is,

A=2[120πr2dθ]

That is,

A=2[120π(2+cosθ)2dθ]=0π(4+2cosθ+cos2θ)dθ=40πdθ+20πcosθdθ+0πcos2θdθ

Use the identity: cos2θ=1+cos2θ2

Thus,

A=40πdθ+20π

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