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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.

Common interior of r = 2 ( 1 + cos θ ) and r = 2 ( 1 cos θ )

To determine

To calculate: The value of the area of the common interior region of r=2(1+cosθ) and

r=2(1cosθ) and to graph the function by the help of graphing calculator.

Explanation

Given:

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

Formula used:

The area of the polar equation is given by:

A=12αβ[f(θ)]2dθ …… (1)

Where, α and β are limits of the integration.

Calculation:

Consider the polar equations r=2(1+cosθ) and r=2(1cosθ).

Now, draw it on TI83 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=4,Xmax=4,Ymin=3 and Ymax=3

Step 5: Press ENTER to get the graph.

The graph is shown below and the shaded section is the common interior of the Polar

equation.

Consider the polar equations:

r=2(1+cosθ) and r=2(1cosθ)

Solve the equations instantaneously and get;

2(1+cosθ)=2(1cosθ)(1+cosθ)=(1cosθ)cosθ=cosθ2cosθ=0

That gives;

θ=π2,3π2

At θ=π2,

r=2(1+cos(π2))=2

At θ=3π2,

r=2(1+cos(3π2))=2

Hence, the point of intersection of one petal is (2,π2),(2,3π2)

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