(* 2D vector field *)
F={x,0};
(* unit circle *)
q={Cos[t], Sin[t]};
p1=VectorPlot[F,{x,-1,1},{y,-1,1},VectorScaling->True];
p2=ParametricPlot[q,{t,0,2 Pi}];
Show[p1,p2]

(* Preliminary computation *)
Foq=F/.{x->q[[1]],y->q[[2]]}
qt=D[q,t]
qtperp={qt[[2]],-qt[[1]]}

(* line integral *)
Integrate[Dot[Foq,qt],{t,0,2 Pi}]

(* flux integral *)
Integrate[Dot[Foq,qtperp],{t,0,2 Pi}]