   Chapter 10.5, Problem 21E

Chapter
Section
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.Inner loop of   r = 1 + 2 sin θ

To determine

To calculate: The value of the area of the inner loop of polar equation r=1+2sinθ and draw it with the use of graphing utility.

Explanation

Given:

The provided polar equation r=1+2sinθ.

Formula used:

The area of the polar equation is given by;

A=12αβ[f(θ)]2dθ

Where; α and β are the limits of the integration.

Calculation:

Consider the polar equation r=1+2sinθ.

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=3,Xmax=3,Ymin=0.5 and Ymax=3.5

Step 5: Press ENTER button to get the graph.

The graph obtained is:

From the graph, the shaded region is the curve formed from the pole r=0.

So, equate the polar equation to 0 and get;

r=01+2sinθ=02sinθ=1sinθ=12

That gives;

θ=7π6 and θ=11π6

The area of the inner loop is given by integration of the polar equation from θ=7π6 to θ=11π6

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