homework and exercises - Why can I assume the force to be constant in this particular interval?

If I have force, or any function $f(z)$, I was told that I can assume it to be constant only in the interval $dz$.

However, in this case, I had to calculate the work done by the spring force as a function of $y$

Over here, I assumed the spring force, which is a function of its elongation $x$ ($F = -kx$) to be constant in the interval $dy$ and integrated and this gave me the correct answer

I want to know why the error vanished over here. Shouldn't spring force only be constant in the interval $dx$ and not $dy$?

I also want to know, in general, if I have a function, how to decide whether it is constant in some particular interval/in which cases the error will vanish as I take the limit and integrate.

Note: I do know I can assume a function $f(x)$ to be constant in the interval $[x,x+dx)$ while integrating, but over here I've assumed it to be constant in the interval $dy$. I want to know why I can do that and also if I can assume a force/function to be constant in any infinitesimal interval such as $Rdθ$, $dy\over cosϕ$,$dz$ etc.