   Chapter 10.5, Problem 44E

Chapter
Section
Textbook Problem

Finding the Area of a Polar Region Between Two CurvesIn Exercises 37-44, use a graphing utility to graph the polar equations. Find the area of the given region analytically.Inside r = 3 sin θ and outside r = 1 + sin θ

To determine

To calculate: The value of the area inside the polar equation r=3sinθ and outside r=1+sinθ, and also graph it by means of graphing calculator.

Explanation

Given:

The two polar equations are r=3sinθ and r=1+sinθ.

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=3sinθ and r=1+sinθ.

Now, draw it on TI83 calculator and do the following steps:

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=3,Xmax=3,Ymin=0.5 and Ymax=3.5

Step 5: Press ENTER to get the graph.

The graph is obtained is shown below and the shaded region is the required portion.

Calculation:

Consider the polar equations are:

r=3sinθ and r=1+sinθ

Solve the equations at the same time;

3sinθ=1+sinθ2sinθ=1sinθ=12

That gives;

θ=π6,5π6

Because above section is symmetric about y-axis. So, double the area formed by the loop.

The area formed by the outer loop is twice the area formed by the equation r=3sinθ by integrating it from θ=π6 to θ=π2.

So,

Aouter=2[12π/6π/2[3sinθ]2dθ]

The area formed by the inner loop is double the area formed by the equation r=1+sinθ by integrating it from θ=π6 to θ=π2.

So,

Aouter=2[12π/6π/2[1+sinθ]2dθ]

So, area inside the polar equation r=3sinθ and outside the equation r=1+sinθ will be the area of shaded section

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