   Chapter 10.5, Problem 38E

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 = 5 − 3 sin θ and r = 5 − 3 cos θ

To determine

To calculate: The value of the area of the common interior section of r=53sinθ and

r=53cosθ and also graph the function by means of graphing calculator.

Explanation

Given:

The provided two polar equations are r=53sinθ and r=53cosθ.

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=53sinθ and r=53cosθ.

Now, draw it on TI83 calculator. 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=12,Xmax=12,Ymin=8 and Ymax=8

Step 5: Press ENTER to get the graph.

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

Calculation:

Consider the polar equations: r=53sinθ and r=53cosθ.

Solve the equations instantaneously;

53sinθ=53cosθ3sinθ=3cosθtanθ=1

This gives;

θ=π4,5π4

So, the area of the section which is interior of curve is given as:

A=2[12π45π4(53sinθ)2dθ]=π45π4

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