I'm trying to understand Green's Theorem and its applications and there's something that just doesn't make sense to me.
Consider the following shape and parametrization:
Why can't I simply use a straight line from (-2,0) to (2,0) to create the simple closed curve required for the theorem? I've seen this working before when dealing with semi-circle regions without the line at the middle (in that case you take the entire integral and then substract the line). Why does it need to be a semi-circle?? I get this at regions with holes at the middle, but a line should do (it actually doesn't but I don't get why)
On the same topic, consider the curve
I will assume this is a simple closed curve because of the drawing. The field is classical in Green Theorem and it's such that Qx - Py = 0 then we only need the little circle in the middle. Some issues with that though:
The small circle's parametrization. According to the solution sheet it's (-sint,cost). Why not use the typical r(t) = (cost, sint)?
(0,0) is actually inside the domain of the function and its image is 0. Why use Green's Theorem then, if the piecewise already took care of that?
Thanks! I'll update this if I'm able to come up with possible explanations, but I'm currently stuck.