   Chapter 10.5, Problem 35E

Chapter
Section
Textbook Problem

# Finding the Area of a Polar Region Between Two Curves In 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 2 θ and r = 2

To determine

To calculate: The area of the common interior region of of the curves r=4sin2θ and r=2 and to graph the function by means of graphing calculator.

Explanation

Given:

The provided two polar equations are r=4sin2θ 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=4sin2θ and r=2.

Now, draw these two curves with the help of TI-83 calculator and do the following steps as shown 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

Xmin=6,Xmax=6,Ymin=4 and Ymax=4

Step 5. Press ENTER button to get the graph.

The graph of the curves is shown below and the shaded region is the common interior of Polar equations.

Calculation:

Consider the polar equations: r=4sin2θ and r=2.

Solve the equations instantaneously;

4sin2θ=2sin2θ=12

That gives;

2θ=π6 and 2θ=5π6

So,

θ=π12 and θ=5π12

Hence, the point of intersections of one petal are (2,π12),(2,5π12).

So, the area of the one petal is;

A=12αβ[f(θ)]2dθ=12[0π12[4sin2θ]2dθ+π125π1222dθ+

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