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In old mathematics books, I see a lot of notations like $\int_{0}^{x} f(x) dx$. For example, Courant-Hilbert: Methods of mathematical physics. However, when I wrote it in this site, it was sometimes edited like $\int_{0}^{x} f(t) dt$.

Makoto Kato
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    It shouldn’t just be frowned on: $\int_0^x f(x)dx$ is simply wrong. – Brian M. Scott Sep 02 '12 at 21:58
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    Although the notation $\int_0^x f(x),dx$ is fine from a logical point of view, I have always avoided it in teaching. Just as I would never write $\forall x(F(x)\land\forall x G(x))$. – André Nicolas Sep 02 '12 at 21:58
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    In most cases it's kind of a nit-picky thing. "Everyone knows" what is meant by $\int_0^x f(x),dx$, but there is something unsatisfactory about $x$ pulling double duty. If a student uses the same variable as a dummy variable and as a limit, I tend to overlook it. But in a textbook, it seems kind of shoddy. Why not use a different symbol for total clarity? It's not like you're going to pay extra for using a different letter. – Bill Cook Sep 02 '12 at 21:59
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    Sometimes I wonder why we don't just write it as $\int_0^x f$. –  Sep 02 '12 at 22:12
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    @RahulNarain : If I write "This leads us to consider the integral $\int_{-\infty}^x \exp(-u^2/2),du$.", I can think of some reasons why I don't just write what you suggest. – Michael Hardy Sep 02 '12 at 22:15
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    @Rahul: We sometimes do. But dummy variable notation is too convenient to discard; e.g. it lets us write things like $\int_0^x t^2 , dt$ without great pain. Sure, you could define $f(t) := t^2$ first so you can write $\int_0^x f$, but you haven't gotten rid of the dummy variable, just shifted it around. –  Sep 02 '12 at 22:30
  • @BrianM.Scott Could you explain why it's simply wrong? – Makoto Kato Sep 02 '12 at 22:31
  • @Makoto: Mathematical grammar is often designed to forbid such ambiguity: you're not allowed to introduce $x$ as a dummy variable into any context where $x$ already appears. –  Sep 02 '12 at 22:35
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    Because it violates the very strong notational convention against using one variable name to refer to two different variables in the same expression. It also subverts one of the main functions of mathematical notation, which is to facilitate understanding. – Brian M. Scott Sep 02 '12 at 22:36
  • @BrianM.Scott As I said, it was used often in old mathematics books. So it's rather traditional notational convention. – Makoto Kato Sep 02 '12 at 22:45
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    It might be a traditional notational convention, but that does not prevent it from being simply wrong by modern standards. – Old John Sep 02 '12 at 23:06
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    @Michael: Indeed, what we need is more pointfree style in mathematics. $\int_{-\infty}^x \exp\circ(-\frac12)\circ\operatorname{square}$, right? :) –  Sep 03 '12 at 01:37
  • @RahulNarain : I suspect I can think of some reasons to avoid your point-free notation, but it's going to take more work to express it cogently. One reason would be that if $f(u)$ is in meters per second and $du$ in seconds, then in the expression $f(u),du$, the seconds cancel and we get the right units. But that's not the only reason. – Michael Hardy Sep 03 '12 at 02:32
  • @Michael, I know, I know, I'm just fooling about at this point. –  Sep 03 '12 at 02:40
  • @Makato Because basically in that case your variable $x$ runs from $0$ to $x$. – N. S. Sep 03 '12 at 04:13
  • @AndréNicolas where is $\forall x(F(x)\land\forall x G(x))$ used?, or in which context? I really have no idea what you are refering to, because of my lack of knowledge. – yiyi Sep 03 '12 at 04:23
  • @MaoYiyi: If you are not familiar with notation from symbolic logic, that part of the comment would not mean much to you. It is an example of using $x$ for two different purposes in the same sentence. It turns out that by the rules of logic it is legal. But as a practical matter it is not a good idea. – André Nicolas Sep 03 '12 at 08:48
  • @BrianM.Scott I don't think just saying it's simply wrong is very persuasive. – Makoto Kato Sep 03 '12 at 09:28
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    I gave you a reason: it violates the currently accepted notational conventions. – Brian M. Scott Sep 03 '12 at 09:45
  • You may also see $a-x.b-x$ in some old books for $(a-x)(b-x)$; I think that I remember seeing it in Cayley’s An Elementary Treatise on Elliptic Integrals, for instance. That doesn’t change the fact that it’s simply wrong today. Notational standards, like standards of rigor, can change over time. – Brian M. Scott Sep 03 '12 at 09:58
  • @BrianM.Scott I'm talking about the 20th century mathematics. It's not that old. – Makoto Kato Sep 03 '12 at 10:04
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    I started reading calculus texts in the late 1950s. I have never seen expressions like $\int_0^xf(x),dx$ in such texts. I doubt that this confusing notation has been in common use in the last $60$ years, at least in the U.S. And it really doesn’t matter: by currently accepted standards it’s at best very confusing and at worst simply wrong. If you understand this fact, I don’t see what your point is. If you don’t, you need to learn it. – Brian M. Scott Sep 03 '12 at 10:15
  • @BrianM.Scott As I wrote, Courant-Hilbert's book(1953) adopted such notations. Tenenbaum-Pollard's Ordinary differential equations(1963) also adopted such notations. – Makoto Kato Sep 03 '12 at 10:55
  • @BrianM.Scott H.M.Edwards Advanced calculus(1993) p.250 – Makoto Kato Sep 03 '12 at 11:26
  • @AndréNicolas thanks for letting me know the branch of math. – yiyi Sep 04 '12 at 00:18
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    @OldJohn Could you explain why it's simply wrong by modern standards? It seems to me that just saying so without an explanation is not very persuasive. – Makoto Kato Sep 24 '12 at 00:20

5 Answers5

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Over the summer I came up with an exercise for the kind of people who like to write $\int_0^x f(x) \, dx$: evaluate

$$\int_1^x \int_x^{x^2} \int_{x^2}^{x^3} x^4 x^5 x^6 \, dx \, dx \, dx.$$

I hope that my point is clear.

Qiaochu Yuan
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  • Ouch! :-) I love it. – Brian M. Scott Sep 03 '12 at 09:46
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    Once upon a time I was a student in a course in which the professor wrote a rather complicated proof on the board, and in one section of the proof used the letter $j$ to refer to one thing, and in another to another. When I mentioned this, she said "Pretend one of them is a Greek $j$." – Michael Hardy Sep 03 '12 at 12:58
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    Here is a case where multiplication is not commutative: compare this with $\int_1^x \int_x^{x^2} \int_{x^2}^{x^3} x^4 , dx, x^5 , dx, x^6 , dx.$ – Ross Millikan Sep 03 '12 at 21:04
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    This doesn't really generalize the notation $\int_{0}^xf(x)dx$. A better generalization would be $$\int_1^x \int_x^{x^2} \int_{x^2}^{x^3} x^4 y^5 z^6 , dx , dy , dz$$ or some such. As far as I can tell, this isn't ambiguous. – goblin GONE Oct 11 '13 at 17:23
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Note that there are two different $x$'s in $\int_{0}^{x} f(x) dx$, which is made explicit when one is changed to $t$. One is the upper limit of integration, which is still free, and the other is the dummy variable bound inside the integral. On careful reading one can tell them apart, but it is easier on the reader and less mistake-prone to distinguish them.

Ross Millikan
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5

This is wrong because it would not allow (without confusion) something like this:

$$\int_0^x f(t, x)\ dt$$

The variable of integration (or summation when doing sums) should always differ from all other variables because, to use expressions I recall (possibly incorrectly) from my youth, the other variables are "bound" (the $x$ above) and the variable of integration is "free", so that the expression is unchanged if the variable is replaced by another.

For example, what would you make of this:

$$\int_0^x \int_0^x f(x,x)\ dx\ dx$$

instead of this:

$$\int_0^u \int_0^y f(x,y)\ dx\ dy$$

marty cohen
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5

If you write $\int_0^x f(x)\,dx$, you have two different $x$s. One has scope inside of the integral, the other outside. The term "scope" is somewhat strange in the mathematical world, but it means "where the variable has meaning."

In this case it is far better to write $\int_0^x f(t)\,dt$; the variable $t$ is a "loop variable" or place-holder.

ncmathsadist
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$\int_0^x f(t) \,dt$

is an expression that, by its limit definition, is basically the sum of an infinite number of areas of rectangular pieces of infinitesimally small width and height $f(t)$ for each value of $t$ between $0$ and $x$. So, if we wrote

$\int_0^x f(x) \,dx$

instead, it would mean to add up the areas of these rectangles as the value of $x$ ranges from $0$ to $x$. Hopefully, it is clear that this makes no sense. $x$ can not simultaneously vary and stay constant.

GeoffDS
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