(* We compute the line integral of F on the upper semicircle with radius 2 in two ways. *)

(* direct calculation *)
R=2
F={y,2x-y}
(* parametrization of the semicircle *)
r={R Cos[t],R Sin[t]}
(* F(r(t)) *)
pF=F/.{x->r[[1]],y->r[[2]]}
D[r,t]
(* F(r(t)).r'(t) *)
g=Dot[pF,D[r,t]]
(* line integral *)
li1=Integrate[g,{t,0,Pi}]

(* Using Green's Theorem *)
(* parametrization of the line on the bottom *)
s={t,0}
(* F(s(t)) *)
pF=F/.{x->s[[1]],y->s[[2]]}
D[s,t]
(* F(s(t)).s'(t) *)
g=Dot[pF,D[s,t]]
(* line integral *)
li2=Integrate[g,{t,-R,R}]
Curl[F,{x,y}]
(* double integral of curl on the interior *)
di=R^2Pi/2Curl[F,{x,y}]
(* line integral on r = double integral of curl - line integral on s *)
di-li2