Simply I saw a friend asking about the area inside the cardoid $r=2+2\sin \theta$ and outside the circle $r=1$ and I couldn't help.
I know that the area is equal to $$
\int_{a}^{b} \frac{1}{2}((2+2\sin \theta)^2-1) \,d\theta
$$
But when I tried to solve the equation $$
2+2\sin \theta =1
$$
I found $\sin \theta =-\frac{1}{2}$
Which means $\theta=-\pi/6$.
Now I'm not quite sure should $a$ be equal to $-\pi/6$ or $7\pi/6$?, and for $b$ should it be $5\pi/6$.
Its so confusing for me because I didn't expect negative sin.
Best Answer
You can plot the graph to guess the values. I think you should integrate from $-\dfrac{\pi}{6}$ to $\dfrac{7\pi}{6}$.