To find: The area of the common region enclosed by the circle
Answer to Problem 53E
The area of the region enclosed by the given circle and the cardioid is
Explanation of Solution
Given information: Equation of circle is
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 using the determined results.
Figure (1)
From the above graph it can be observed that the common region enclosed by the given circle and cardioid is same as the region enclosed by the polar curve
Also, the common region enclosed by the given circle and cardioid is symmetrical with respect to the
Solve the integral to find the required area.
Further solve the integral.
Apply the trigonometric identity
Further solve the integral.
Evaluate the integral
For
Use the above results determine the area.
Therefore, the area of the region enclosed by the given circle and the cardioid is
Chapter 10 Solutions
AP CALCULUS TEST PREP-WORKBOOK
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