What is $dx$ in integration?

Question

When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board.

$$\int f(x)\, dx$$

When he came to explain the meaning of the $dx$, he told us "think of it as a full stop". For whatever reason I did not raise my hand and question him about it. But I have always shaken my head at such a poor explanation for putting a $dx$ at the end of integration equations such as these. To this day I do not know the purpose of the $dx$. Can someone explain this to me without resorting to grammatical metaphors?

Answer 1

The motivation behind integration is to find the area under a curve. You do this, schematically, by breaking up the interval $[a, b]$ into little regions of width $\Delta x$ and adding up the areas of the resulting rectangles. Here's an illustration from Wikipedia:

Then we want to make an identification along the lines of

$$\sum_x f(x)\Delta x\approx\int_a^b f(x)\,dx,$$

where we take those rectangle widths to be vanishingly small and refer to them as $dx$.

Answer 2

There are multiple ways of explaing what the $dx$ means.

• Practical explanation: It says we are integrating over variable $x$. If we were to integrate over variable $t$, we would write $dt$ instead, and so on.

• Infinitesimal explanation: We can think of an integral as the limit of a sum: The area under the graph of a (positive) function $f$ can be approximated by the sum $\sum_x f(x) \Delta x$, and in the limit, we make $\Delta x$ arbitrarily small and call it $dx$ (an "infinitesimal" quantity). Jonathan's answer explain that in detail.

• Advanced explanation: In vector analysis, $dx$ takes meaning as a differential form (roughly, something that behaves like an infinitesimaly small piece of a curve).

Answer 3

Can someone explain this to me without resorting to grammatical metaphors?

It is a matter of grammar. The indefinite integral expression is a large expression organizing several pieces of information:

$$ \color{blue}\int \color{red}{\underline{\quad}} \color{green}d \color{purple}{\underline{\quad}} $$

The blue $\int$ is a symbol expressing that this is an integral expression. The rest of the expression is the integrand.

The integrand consists of three components: there is the green $d$ symbol. There is the purple slot on the right in which you place the name of the variable that you are integrating with respect to, and there is the red slot on the left in which you place the function expression you intend to integrate (with respect to the dummy variable).

There are other grammatical interpretations of integral expressions — most importantly (IMO) the notion of a "differential form" — but this is the one you are using in your introductory calculus class.

This particular grammatical form has some symbolism. It is a useful heuristic to think of a "$dx$" as a miniature variation in a function. You can extend this heuristic by imagining the integral to be "adding" up all of these miniature variations. The symbol $\int$, I believe, originated as an elongated $S$, for "sum"; not dissimilar to the choice of sigma ($\Sigma$) for summation expressions.

The notion of differential form is a very useful one you may be interested in learning more about. Unfortunately, I am not aware of any exposition that introduces it as applied to introductory calculus: it's usually only really introduced in a differential geometry course.

Answer 4

Leibniz, who introduced this notation in the 17th century, thought of $dx$ as an infinitely small increment of $x$, and at least as a heuristic, that is an immensely useful idea.

However, note some other points:

• $\displaystyle\int f(x,y)\,dx$ differs from $\displaystyle\int f(x,y)\,dy$. In one case, one integrates a function of $x$, and $y$ is constant; in the other these roles are reversed and one might be integrating a very different function.
• If $f(x)$ is in meters per second and $dx$ is in seconds, then $f(x)\,dx$ is in meters, and so is the integral. These things should be dimensionally correct, and are not so without the "$dx$".
• Sometimes one has a dot-product or a cross-product or a matrix product or some other sort of product between $f(x)$ and $dx$. How would one specify that without the "$dx$" written there?
• When doing substitutions, it becomes important to distinguish between $dx$ and $du$, etc.

Answer 5

The $dx$ can be given various concrete meanings, none of which one can sensibly explain to someone first learning about integrals. It is, in reality, just a notation which comes to use from the originators of calculus, motivated by the ideas behind Jonathan's answer.

Today, the $dx$ serves the purpose of delimiting the integrand (although the physicists, rebellious as ever, like to write $\int\mathrm d xf(x)$ for what we write $\int f(x)\mathrm dx$...) and of making explicit the variable respect to which we are computing the integral (this s useful in situations like $\int f(x,y)\mathrm dx$, which is usually different from $\int f(x,y)\mathrm d y$)

As for concrete mathematical meanings: the $\mathrm dx$ can mean concretely all sort of things: the Lebesgue measure, a differential form, a density, and a few others. It would be impossible to explain what any of these mean to a student first encountering integrals!

Answer 6

Historically, calculus was framed in terms of infinitesimally small numbers. The Leibniz notation dy/dx was originally intended to mean, literally, the division of two infinitesimals. The Leibniz notation $\int f dx$ was meant to indicate a sum of infinitely many rectangles, each with infinitesimal width dx. (The integral sign $\int$ is an "S" for "sum.") Note that the factor $dx$ in the integral is needed in order to make the units come out right. For example, if you're calculating mechanical work as $W=\int F dx$, the units wouldn't be newton-meters if you didn't have the factor of $dx$, which has units of meters.

In the 19th century, mathematicians got uneasy about infinitesimals. They were afraid that a system of mathematics based on infinitesimals could not be developed in a fully rigorous and consistent way. Therefore, they rebuilt the foundations of calculus using limits, but they kept the Leibniz notation, which is extremely useful and practical. In this approach, $W=\int F dx$ stands for a limit of Riemann sums of rectangles with finite widths $\Delta x$, and the $dx$ becomes an archaism.

Around 1960, Abraham Robinson showed that it was possible to have calculus built on a foundation of infinitesimals, and that no inconsistency would result (unless there was an inconsistency that would also affect the real number system itself, which nobody thinks is the case). Therefore it's legitimate to think of integrals and derivatives in essentially the same way that Newton and Leibniz originally conceived of them -- in fact, scientists and engineers never actually stopped thinking about them that way.

Answer 7

I once went at some length illustrating the point that for the purpose of evaluating integrals it is useful to look at $d$ as a linear operator.

Answer 8

Of course for something as simple as $\int{f(x)}dx$, you dont have to write $dx$ if you don't feel like it, and in many situations you are allowed to just write $\int{f}$, although I don't personally make a habit of it.

These things you ask about are not merely some convenient book-keeping device to let us know where the end of the intergral is, they are called differential forms, and you can add and multiply them together.

The algebra of differential forms follow naturally from the simple rule that $dx^2=0$ because this rule actually implies another very important rule, namely that $dx\wedge dy=-dy\wedge dx$, or in other words, that differential forms commute anti-symmetrically, see here for more info.