Help! Find the area of the inner loop of the cardiod r= 1+ 2*sin (t)?

Question

Help! Find the area of the inner loop of the cardiod r= 1+ 2*sin (t)?

Answer 1

wow what is that?

Answer 2

7 * pi

My book lists the parameters in cartesian coordinates, not polar coordinates, but I figured the transformation in my head -- not on paper -- so I am only 99% certain of the answer.

The Area given in the book is: 7 * pi * a^2

where: a is the radius of the two circles that form the cardiod.

In this case it is obvious that a=1.  

If the radius of the 'circling' circle is anything other than 1, then it would be a general epiclycloid and not a cardiod.  For a cardiod, the 2 circles must have the same radius.

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