So I have this vector field $$V(x,y)=(xy,x+y)$$ that I am calculating its circulation with 2 methods (Using a parametrization and using Green's theorem), The domain we're working on it is $${(x,y)\in R^2; x^2+y^2\le4 ; 0\le x \le y/\sqrt3 }$$
so its circulation is decomposed into 3 circulations: (The first method)
$$C_{1}: \text{from} \; O(0,0)\; \text{to} \; B(1,\sqrt3); \; \text{set} \; x=t, y=t\sqrt3, 0\le t \le 1 $$
$$C_{2}: \text{from} \; B(1,\sqrt3)\; \text{to} \; C(0,2); \; \text{set} \; x=2\cos t, y=2\sin t, \pi/3\le t \le \pi/2 $$ $$C_{3}: \text{from} \; C(0,2)\; \text{to} \; O(0,0); \; \text{set} \; x=0, y=t, 2\le t \le 0 $$ I want to know how did we find the values of x and y on each circulation, is there a method to do so in different cases? and how can I tell the limits of "t" in every circulation (specially the one in $C_{3}$ how is that even possible ?) in the second method (Green's method) $$C= \int_{0}^{1} \int_{x\sqrt3}^{\sqrt(4-x^2)}(1-x) dydx$$ Here I am just confused on how did we get those integration limits? Hope everything is clear, Thanks!