Chapter 10.5, Problem 41E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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=4sinÎ¸Â andÂ r=2.

Formula used:

The area of the polar equation is given by;

A=12âˆ«Î±Î²[f(Î¸)]2dÎ¸

Where Î±Â andÂ Î² are limits of the integration.

Calculation:

Consider the polar equations r=4sinÎ¸Â andÂ 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=âˆ’6,Xmax=6,Ymin=âˆ’3Â andÂ 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=4sinÎ¸Â andÂ r=2

Solve the equations of the curves instantaneously;

4sinÎ¸=2sinÎ¸=12

That gives;

Î¸=Ï€6,5Ï€6

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=4sinÎ¸ from Î¸=0Â toÂ Î¸=Ï€6 and the curve r=2 from Î¸=Ï€6Â toÂ Î¸=Ï€2

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