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I know this question has a little bit of physics in it, but it's mostly about maths:

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 $y$ enter image description here

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.

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    I don't really think this is about mathematics; What your question centers around is essentially how work is computed, so it's about physical definition(?) or derivation of the work=integral of force over distance – 2017-02-19
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    I think it is about mathematics. There is no "error" because acccording to the definition of work, you are passing from a Riemann sum (whence the "constant on dz" abuse of terminology), to the integral. That is, each Riemann sum may be considered an approximation of the work done, which becomes exact upon taking the appropriate limit. – 2017-02-19
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    +chilango incomprendido I 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 $Rd\theta$, $dy/cos\phi$, $dz$ etc. – 2017-02-19

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