Deconstructing dx

May 5, 2008, 1:48 pm

Asking the following question may make me less of a mathematician in some people’s eyes, and I’m fine with that, but: How do you explain the meaning of the differential dx inside an integral? And more importantly, how do you treat the dx in an integral so that, when you get to u-substitutions, all the substituting with du and dx and so on means more than just a mindless crunching of symbols? 

Here’s how Stewart’s Calculus does it: 

  • In the section introducing the definite integral and its notation, it says: “The symbol dx has no official meaning by itself; \(\int_a^b f(x) \, dx\) is all one symbol.” (What kind of statement is that? If dx has “no official meaning”, then why is it there at all?) 
  • In the section on Indefinite Integrals and the Net Change Theorem, there is a note — almost an afterthought — on units at the very end, where there is an implied connection between \(\Delta t\) in the Riemann sum and dt in the integral, in the context of determining the units of an integral. But no explicit connection, such as “dx is the limit of \(\Delta x\) as n increases without bound” or something like that. 
  • Then we get to the section on u-substitution, which opens with considering the calculation of \(\int 2x \sqrt{x^2+1} \, dx\) (labelled as (1) in the book). We get this, er, explanation: 

Suppose that we let u be the quantity under the root sign in (1),  \(u = 1 + x^2\). Then the differential of u is du = 2x dx. Notice that if the dx in the notation for an integral were to be interpreted as a differential, then the differential 2x dx would occur in  (1), and, so, formally, without justifying our calculation, we could write \(\int 2x \sqrt{1+x^2} \, dx = \int \sqrt{u} \, du\)…

So, according to Stewart, dx has “no official meaning”. But if we were to interpret dx as a differential — he makes it sound like we have a choice! — then using purely formal calculations which we will not stoop to justify, we could write the du in terms of dx. That is, integrals contain these meaningless symbols which, although they have no meaning, we must give them some meaning — and in one particular way — or else we can’t solve the integral using these purely formal and highly subjunctive symbolic manipulations that end up getting the right answer. 

Er, right. 

To be fair, my usual way of handling things isn’t much better. I start by reminding students of the Leibniz notation for differentiation, i.e. the derivative of y with respect to x is dy/dx. Then I say that, although that notation is not really a fraction, it comes from a fraction — and that much is true, since dy/dx is the limit of \(\Delta y / \Delta x\) as the interval length goes to 0 — and so we can treat it like a fraction in the sense that, say, if \(u = x^2 + 1\) then \(du/dx = 2x\) and so, “multiplying by dx”, we get \(du = 2x dx\). But that’s not much less hand-wavy than Stewart. 

Can somebody offer up an explanation of the manipulation of dx that makes sense to a freshman, works, and has the added benefit of actually being true? 

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