|Module 25 - Polar Functions|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|Lesson 25.3: Area Bounded by Polar Graphs|
Lesson 25.1 introduced polar coordinates and Lesson 25.2 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
The definite integral can be used to find the area of the region enclosed by the cardioid
The entire graph of this function is plotted for , so the area is given by the definite integral
Evaluate this integral on your TI-89.
The area enclosed by the cardioid is 6 square units.
25.3.1 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 is given by
The definite integral can be used to find the area that lies inside the circle r = 1 and outside the cardioid r = 1 cos .
First visualize 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 symbol is entered by pressing .
The definite integral that gives the area is
The area is square units.
25.3.2 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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