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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Section
BuyFindarrow_forward

Calculus

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

Finding the Area of a Polar Region In Exercises 19-26, use a graphing utility to graph the polar equation. Find the area of the given region analytically.

Between the loops of   r = 1 2 + cos θ

To determine

To calculate: The value of the area between the loops of polar equation r=12+cosθ and draw the area by the use of graphing utility.

Explanation

Given:

The polar equation is r=12+cosθ.

Formula used:

The area of the polar equation is A=12αβ[f(θ)]2dθ.

Where, α and β are limits of the integration.

Calculation:

Consider the polar equation r=12+cosθ.

Now, use the following steps in the TI-83 calculator to obtain the graph:

Step 1: Press ON button 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 button and then set the window as follows:

Xmin=1,Xmax=2,Ymin=1 and Ymax=1

Step 5: Press ENTER to get the graph.

The graph obtained is:

Since, it can be seen that there is symmetry in above graph.

So, the area inside the outer loop is twice the area obtained by the integration of the polar equation r=12+cosθ from r=0 to r=32.

At r=0, the value of θ is:

0=12+cosθ12=cosθ

This gives;

θ=2π3 and θ=4π3

And, at r=32, the value of θ is:

32=12+cosθ3212=cosθ1=cosθ

This gives;

θ=0

So, to get the area inside the outer loop, twice the area obtained by the integration of the polar equation r=12+cosθ from θ=0 to θ=2π3.

So, the area inside the outer loop is:

Aouter=2[1202π/3[f(θ)]2dθ]=02π/3[12+cosθ]2dθ=02π/3[14+cos2θ+cosθ]dθ

Use the identity 1+cos2θ=2cos2θ and get;

Aouter=02π/3[14+12(1+cos2θ)+cosθ]dθ=02π/3[34+12cos2θ+cosθ]dθ=[34θ+14sin2θ+sinθ]02π/3

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