I am more than happy understanding how to find the equation for the electric field for a ring of charge along its centre axis (see [Electric Field: Ring of Charge](http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html#c2)). But in every explanation I have seen they use the symmetry of the problem to cancel out one of the components. I understand why they do this, it makes the calculations much easier, but I would like to understand how mathematically it cancels out. Any help would be appreciated. Thank you.