Undergrad Limits of integration on Polar curves

[CrazyNeutrino]

General question, how do you determine the limits of integration of a polar curve? Always found this somewhat confusing and can't seem to find a decent explanation on the internet.

 

[BvU]

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

 

[CrazyNeutrino]

Oh sorry, any closed curve defined in Polar coordinates. Cardiods, limacons, circles, the works.

 

[BvU]

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 ?

 

[CrazyNeutrino]

The area enclosed by the Polar curve using Int(1/2 r^2) d theta. I find the determination of the limits of integration slightly ambiguous when I watch any tutorials or read up on Polar coordinates. I normally just use graph trace but I'd like to get an intuitive understanding

 

[BvU]

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:

normally just use graph trace

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.

I find the determination of the limits of integration slightly ambiguous

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.