   Chapter 10.5, Problem 43E

Chapter
Section
Textbook Problem

# Finding the Area of a Polar Region Between Two CurvesIn Exercises 45-48, find the area of the region.Inside r = a ( 1 + cos θ ) and outside r = a cos θ

To determine

To calculate: The value of the area inside the polar equation r=a(1+cosθ) and outside r=acosθ and to graph it by means of graphing calculator.

Explanation

Given:

The two polar equations are r=a(1+cosθ) and r=acosθ.

Formula used:

The area of the polar equation is given by;

A=12αβ[f(θ)]2dθ

Where α and β are limits of the integration.

Calculation:

Consider the polar equations r=a(1+cosθ) and r=acosθ.

Let assume a=2. So, the polar equations become r=2(1+cosθ) and r=2cosθ

Now, draw it on TI83 calculator. Do the following steeps:

Step 1: Press ON to open the calculator.

Step 2: Press MODE button and then scroll down to press pol and press ENTER button.

Step 3: Now, press the button Y= and enter the provided equation.

Step 4: Press WINDOW and then set the window Xmin=5,Xmax=5,Ymin=5 and Ymax=5

Step 5: Press ENTER to get the graph.

The graph is shown below and the shaded region is the required portion.

Consider the polar equations:

r=a(1+cosθ) and r=acosθ

Since the graph is symmetric to x-axis. So, to get the area of the outer loop, double the area obtained by integrating the outer loop r=a(1+cosθ) from θ=0 to θ=π.

So, the value of the area inside the outer loop is;

Aouter=2[120π[a+acosθ]2dθ]

The area of inner loop is obtained by integrating r=acosθ from θ=π2 to θ=3π2.

Ainner=2[12π/23π/2[acosθ]2dθ]=π/23π/2a2cos2θdθ=π/23π/2a2(1+cos2θ2)dθ

Further simplify and get,

Ainner=[a22θ+a24(sin2θ)]π/2

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