Module 27  Polar Functions  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 27.3: Area Bounded by Polar Graphs  
Lesson 27.1 introduced polar coordinates and Lesson 27.2 investigated the graphs of polar equations. This lesson explores finding the area bounded by polar graphs. Finding the Area Bounded by Polar Graphs The area of the region between the origin and the curve r = f( ) for is given by the definite integral
This definite integral can be used to find the area of the region enclosed by the cardioid The complete graph of this function is plotted using the interval , so the area is given by the definite integral The integral may be evaluated on your TI83.
The area enclosed by the cardioid is approximately 18.8496 square units. The exact area is 6 square units. 27.3.1 Find the area enclosed by the curve r = 2 on the interval . Click here for the answer. Finding the Area Between Two Polar Curves The area bounded by two polar curves where on the interval is given by This definite integral can be used to find the area that lies inside the circle r = 1 and outside the cardioid r = 1 – cos . First illustrate the area by graphing both curves.
The area inside the circle and outside the cardioid lies in the first and fourth quadrants. To find the area between the curves you need to know the points of intersection of the curves. The TI83 can display coordinates in Polar form.
The first point of intersection occurs when .
The second point of intersection occurs when .
The two points of intersection are at
and
. However, the values of
in the interval
correspond to the parts of the graphs that are in quadrants II and III. We want the area bounded by the graphs in quadrants IV and I. In order to obtain the proper limits of integration notice that the polar point
is the same as
so the desired area is found by letting
range from The definite integral that gives the area is
The area is approximately 1.215 square units. 27.3.2 Find the area of the region inside the circle r = 3 sin and outside the cardoid r = 1 + sin . Click here for the answer. 

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