integration – What is the geometric interpretation of a unit circle having a line integral of zero along a curve?

If F(x,y) = x^2 +y^2 and has a curve C that is an arbitrary arc, what would it mean in a geometric sense for it to have a line integral of 0? I thought it would suggest that the curve must go along the entirety of the circle to make sure it has the same start and end points. Is this right?