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When we integrate something say work, $\int F\cdot ds $ then we will get work but what exactly is $ds$? how much is ds? Is it the Planck length? Are we just pretending there exists some infinitesimals and all the math works out in the end?

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closed as unclear what you're asking by my2cts, StephenG, AccidentalFourierTransform, ZeroTheHero, Qmechanic Jul 20 at 18:23

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    $\begingroup$ Planck length would suffice for most cases. If in doubt, use half a Planck length. You may have a run time problem though. $\endgroup$ – my2cts Jul 20 at 9:26
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    $\begingroup$ Well, you are integrating a function which represents a model of Nature's behaviour, so as soon as you accept it as a valid model and solve the integral there can be two outcomes: either there is an analytical solution ($ds \rightarrow 0$) or you have to solve it numerically, in which case you'll have to sample the integrand to discrete points by defining a $\Delta s$, that you'll want to approximate to zero as close as possible. At the end of the day, it doesn't really matter, since we can't measure anything with infinite resolution. $\endgroup$ – Lith Jul 20 at 9:45
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    $\begingroup$ Are you happy with the standard approach to defining the real numbers and the Riemann integral (so you've done the equivalent of a course on real analysis). If that's not the case then this question probably belongs on the maths stack exchange. $\endgroup$ – tfb Jul 20 at 10:26
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    $\begingroup$ @Lith That comment would probably make a good answer. $\endgroup$ – StephenG Jul 20 at 10:38
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    $\begingroup$ @Lith Please can you convert your comment into an answer? I agree with StephenG $\endgroup$ – Sebastiano Jul 20 at 11:51
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Integration is purely a mathematical process.

Infinitesimal work $\delta W$ along a path $\vec dl$ in space is given by $\delta W = \vec F \cdot \vec dl$. When you integrate the work among a certain path, you use usually the Riemann integration, however, other definition of integration exist. Riemann integration is a sum of infinitesimal values of the function in a certain range. For the case of work given by the formula above, you integrate $\vec dl$ over a certain path. $\vec dl$ has no physical reality, it is only a mathematical concept. However, the path to which you integrate has a sense, it is the real path calculated.

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As @PackSciences has stated in his answer, the symbol $ds$ simply denote an infinitesimal, in the sense that it tells us the variable over which we are carrying out the integration. Now, in the lines of my previous comment, there are cases (most of them) where there is no analytical solution to the integral and we must solve it numerically, and there's when we approximate the differential $ds$ by some finite quantity $\Delta s$.

But perhaps you know this and you are referring to the philosophical question about the ultimate nature of reality, that is, is reality discrete or continuous? In that case, you might find interesting the essay contest about this very question undertaken by The Foundational Question Institute some years ago. Check this (and links therein): Is reality digital or analog?

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