To find: The area of the common region enclosed by the cardioids
Answer to Problem 54E
The area of the region enclosed by the given cardioids is
Explanation of Solution
Given information: Equation of the cardioids are
Formula used: The double-angle formula for cosine function is given by:
Calculation:
Equate the given equation of the circle and the cardioid to find the value of
According to the fact
Further simplify.
Draw the graph of the given circle and cardioid.
Figure (1)
From the above graph it can be observed that the common region enclosed by the given cardioids is same as the region enclosed by the polar curve
Also, the common region enclosed by the given cardioids is symmetrical with respect to the
Calculate the required area enclosed by the given curves.
Further simplify the above integral.
Apply the trigonometric identity
Further simplify.
Evaluate the integral
For
Using the above results determine the integral
Therefore, the area of the region enclosed by the given cardioids is
Chapter 10 Solutions
Advanced Placement Calculus Graphical Numerical Algebraic Sixth Edition High School Binding Copyright 2020
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