   Chapter 10.5, Problem 40E

Chapter
Section
Textbook Problem

# Finding the Area of a Polar Region Between Two CurvesIn Exercises 37-44, use a graphing utility to graph the polar equations. Find the area of the given region analytically.Common interior of r = 2 cos θ and r = 2 sin θ

To determine

To calculate: The value of the area of the common interior region of

r=2cosθ and r=2sinθ and to graph the function by the use of graphing calculator.

Explanation

Given:

The two polar equations are r=2cosθ and r=2sinθ.

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=2cosθ and r=2sinθ.

Now, draw it on TI-83 calculator. Use the following steps on TI-83 calculator:

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=2,Xmax=4,Ymin=2 and Ymax=3

Step 5: Press ENTER to get the graph.

The graph is shown below and the shaded region is the common interior of polar equation.

Consider the polar equations are: r=2cosθ and r=2sinθ.

Solve the equations simultaneously and get;

2sinθ=2cosθtanθ=1

This gives;

θ=π4,5π4

To get the area between the curves, twice the area obtained by the integration of the polar

equation r=2cosθ from θ=π4 to θ=π2

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