   Chapter 10.5, Problem 20E

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 = 4 − 6 sin θ

To determine

To calculate: The value of the area of the inner loop of polar equation r=46sinθ and to draw the equation with the graphing utility.

Explanation

Given:

The provided polar equation r=46sinθ.

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:

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=8,Xmax=8,Ymin=12 and Ymax=2

Step 5: Press ENTER to get the graph.

The graph obtained is:

From the graph, it can be seen that there is symmetry in the graph above the y-axis.

So, the shaded region is the twice the region formed from the curve r=2 to r=0.

Now equate the polar equation equal to 0;

That is,

r=046sinθ=06sinθ=4sinθ=23

That gives;

θ=sin1(23)

And, at r=2, the value of θ is;

2=46sinθ2=2sinθ1=sinθ

That gives;

θ=π2

The area of the inner loop is twice the area formed by integration of the polar equation from θ=π2 to θ=arcsin(23)

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