I am answering sample exams for my Calculus class and my attention was caught by the following item.
Set-up the definite integral or sum of definite integrals equal to the area of the region above the polar axis, inside the limaçon $r = 3 + 2 \sin \theta$ and outside the lemniscate $r^2 = 32 \cos 2\theta$ given that the two curves intersect at $(4,\frac{\pi}{6})$.
r = 3 + 2 \sin \theta (blue) and $r^2 = 32 \cos 2\theta$ (violet) from $0$ to $2\pi$ (generated by _Mathematica_)">
At first, I thought that the area is given by $\dfrac{1}{2} \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}}{[(3 + 2\sin \theta)^2 - (32 \cos 2\theta)] \mathrm{d}\theta}$ but I know that the area of the lemniscate is tricky so I may have given a smaller area.
My question is this: How do you know the limits of integration for lemniscates? (I know that the limits of integration for the area of the lemniscate alone is from $-\frac{\pi}{4}$ to $\frac{\pi}{4}$, but how about for small portions of the curve?)
I'll appreciate any help. Thank you so much.