You are reparameterizing in terms of arc length so s(t) is the distance traveled in time t.

In most instances you are at the initial point when t= 0 which is why the bottom limit is 0. you are at the point (1,0,0) at t = 0. It has nothing to do with the spatial coordinates but rather the time coordinate.

If for instance you were at the initial point (1,0,0) at t= t0 then t0 would be the lower limit of integration