   Chapter 10.5, Problem 41E

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.Common interior of r = 4 sin θ and r = 2

To determine

To calculate: The area of the common interior region of the curves r=4sinθ and r=2 and to graph the function by means of graphing calculator.

Explanation

Given:

The provided two polar equations are r=4sinθ and r=2.

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=4sinθ and r=2.

Now, draw these curves on TI-83 calculator and do the following steps as given below:

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=6,Xmax=6,Ymin=3 and Ymax=6

Step 5: Press ENTER to get the graph.

The graph of the two curves is as shown below and the shaded region is the common interior of Polar equation.

Consider the polar equations are: r=4sinθ and r=2

Solve the equations of the curves instantaneously;

4sinθ=2sinθ=12

That gives;

θ=π6,5π6

Since, it can be seen that the graph is symmetrical to y-axis.

So, the area of the region which is interior of curve is twice the area formed by the curve

r=4sinθ from θ=0 to θ=π6 and the curve r=2 from θ=π6 to θ=π2

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