Find the area for one loop of

I plotted three graphs; vs. [tex]\theta]/math], where it looks like a normal sin curve repeating twice between 0 and ranging from 64 to -64. Another of r vs. where it looks kind of like two elliptical shapes ranging from -8 to 8, restricted to 0 to and to . Finally, the polar graph on x and y axis, where it looks like a lemniscate reaching a limit of 8 at and .

I did this:

Solving that I got . I think the answer is , though. Someone suggested I set the interval to . I understand what they're doing... integrating to the maximum r and doubling that, but I don't understand why that gives the correct answer and doing it the other way does not.

I think I may have a couple of different computational errors going on, despite trying and retrying the problem, as well as a conceptual block that I'm just not getting.

Any help, much appreciated.