离散分划和推广为连续积分的合理性及微分来源疑问

Hey there! I totally get why that "obvious" transition feels confusing—textbooks love glossing over the nitty-gritty when they assume you’re following along, but that leap from sums to integrals with a seemingly random differential does need unpacking. Let’s break it down step by step:

Anchor Back to Riemann Sums (Your Known Reference) #

You mentioned you grasp the Riemann sum idea, so let’s use that to connect the dots. Suppose we’re calculating a discrete partition sum for something like total mass over an interval [a,b]:

• Split the interval into n tiny subintervals, each with finite width Δx_i
• For each subinterval, pick a point x_i and approximate the mass in that slice as ρ(x_i) * Δx_i (where ρ is density)
• The total discrete sum is Σ ρ(x_i)Δx_i (summed over all subintervals)

When we take the limit as n→∞ and every Δx_i→0, this finite sum becomes the definite integral ∫ₐᵇ ρ(x) dx.

Where Does the Differential dx Come From? #

The dx in the integral is directly tied to that Δx_i from the Riemann sum. Here’s the key: when moving to the continuous case, we’re no longer dealing with measurable, finite Δx steps—we’re working with infinitesimally small intervals. The notation dx is just how we represent that infinitesimal "step size" in the limit.

Textbooks say "replace the sum with an integral and multiply by the differential" because:

• The discrete sum symbol Σ becomes the integral symbol ∫ (a stylized "S" for sum, representing the infinite sum of infinitesimal terms)
• The finite step Δx becomes the infinitesimal dx (the limit of Δx as it shrinks to zero)

Why Does the Textbook Call It "Obvious"? #

To a mathematician who’s spent years working with these limits, the link between Riemann sums and integrals is second nature. They see the discrete sum as a rough finite approximation of the continuous integral, so swapping Σ for ∫ and Δx for dx feels like a natural extension. But for someone learning this, that leap isn’t obvious at all—you’re totally right to question it!

The "Missing" Step Length #

You noticed the integral expression doesn’t show a step length, and that’s because dx is the stand-in for that infinitesimal step. The integral notation hides the entire limit process (the n→∞ and Δx→0 part) inside the ∫ symbol, so it looks like the step vanished—but it’s actually just been rebranded as dx.

If you wrote the integral explicitly as a limit, it would look like this:
∫ₐᵇ f(x) dx = limₙ→∞ Σᵢ₌₁ⁿ f(x_i)Δx_i
Here you can see the Δx_i clearly, but once we switch to standard integral notation, we condense that entire limit process into ∫ and dx.

内容的提问来源于stack exchange，提问作者B. Brekke