/sci/ - Science & Math

so first of all sorry for the shitty diagram.

I am studying Amperes law and was doing some exercises for myself to try and understand it more intuitively. Here I am trying to reason how
[math]\oint\vec{B}\cdot d\vec{s}=0[/math]
in this situation.

I have convinced myself by symmetry that all dot products of [math]\vec{B}[/math] and [math]d\vec{s}[/math] cancel except for at the points in the center of the paths BC and DA, at which the induced magnetic field by the current is parallel (for BC) or antiparallel (for DA) to the corresponding path element, and which there is no symmetrical elements for either. Furthermore it is obvious [math]||\vec{B}||[/math] at the point marked [math]y[/math] is greater than [math]||\vec{B}||[/math] at the point marked [math]x[/math], so I am not convinced they cancel out.

Do we simply ignore these points or approximate them as equal since they are infinitesimal pieces? I feel like I am missing something with my reasoning.