So I have the following question here which says:
Find the total area enclosed between the large loops and small loops of the curve $r=1+2\sin(3\theta)$.
I have no issues with the computation of the integral which is simply $\displaystyle\frac{1}{2}\int (f(\theta))^2 d\theta$ but I'm having a little trouble finding the bounds of the integral.
I did attempt to find the roots of the function, which are $\frac{7\pi}{18}$,$\frac{11\pi}{18}$,$\frac{19\pi}{18}$,$\frac{23\pi}{18}$ and $\frac{31\pi}{18}$ but at each of these points, the outer, bigger curve doesn't quite make a full loop. In fact, if I start at $\theta=0$, the first loop isn't fully enclosed all the way until $2\pi$.
Thus, I can't also find the bounds of the inner loop either since that's not plotted until much later.
My idea is the just find the area of one "petal" and then just multiply by $3$ but that doesn't seem to be possible...
Can anyone help me setup the bounds?
Thanks!
Hint…
Your list of roots is incomplete. You need to find all the roots and list them in numerical order.
$$-\frac{\pi}{18}, \frac{7\pi}{18}, \frac{11\pi}{18}, \frac{13\pi}{18}…$$
The curve is traced anti-clockwise passing through the origin at each of these values of $\theta$.
Using the first and second as bounds will give you the area of the larger loop on the left.
Using the second and third will give you the smaller loop at the bottom.