I Limits of integration on Polar curves

What's a polar curve ? A circle on the South pole ? A trajectory described in polar coordinates ?
Please give a clearer description.

 

I should have asked straight away too: What is it you want to integrate ? some function over the surface, over the boundary ? A vector function ? Just the circumference or the area ?

 

Browsing some of the links at the lower left might be instructive.
We did a cardioid here not so long ago (no full solution, just hints).
Point is: with a concrete example we can see where things go wrong for you.
As you can see in the link, I am an advocate of your approach:

and from that, with experience, grows intuition. The latter two aren't sold by weight (in contrast with what some managers seem to think).

I don't think I can provide much guidance based on e.g.

All I can say is usually ##\theta## runs from 0 to ##2\pi## or from ##-\pi## to ##+\pi##. But I doubt if that is helpful for you.