I have a cardioid with a trisectrix, making a loop inside. This is what the cardioid looks like:
. My question is the following, if we use Green's theorem to calculate the area of C, which is the cardioid defined as $r(t) = (1 + 2 \cos(t)) \sin(t)\mathbf i + [1 + (1 + 2 \cos(t)) \cos(t)]\mathbf j$, will the area of the loop be added on top of the area of the cardiod when we use t from 0 to pi? The loop starts at $t = 2\pi/3$ and ends at $4\pi/3$. The way I see it, the cardioid will be calculated from $t = 4\pi/3$ to $2\pi/3$, and then the loop will be calculated as well between $t = 2\pi/3$ and $4\pi/3$, which would make the cardioid's area one loop too large. What do you think?