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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Finding the Area of a Polar Region Between Two Curves In Exercises 37-44, use a graphing utility to graph the polar equations. Find the area of the given region analytically.

Inside r = 2 cos θ and outside r = 1

To determine

To calculate: The value of the area inside the polar equation r=2cosθ and outside r=1, and to graph the area by the use of graphing calculator.

Explanation

Given:

The provided two polar equations are r=2cosθ and r=1.

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=1.

Now, draw it on TI-83 calculator. Use the following steps of 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=2.5,Ymin=1.5 and Ymax=1.5.

Step 5: Press ENTER to get the graph.

The graph is shown below and the shaded region is the mutual interior of Polar equation.

Calculation:

Consider the polar equations: r=2cosθ and r=1

Solve the equations at the same time and get;

1=2cosθcosθ=12

This gives;

θ=π3,5π3

The area formed by the outer loop is double the area formed by the equation r=2cosθ by the integration from θ=0 to θ=π3.

So,

Aouter=2[120π/3[2cosθ]2dθ]

The area formed by the inner loop is double the area formed by the equation r=1 by the integration from θ=0 to θ=π3

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