Usage of dx in Integrals

All the integrals I'm familiar with have the form:

$\int f(x)\mathrm{d}x$.

And I understand these as the sum of infinite tiny rectangles with an area of: $f(x_i)\cdot\mathrm{d}x$.

Is it valid to have integrals that do not have a differential, such as $\mathrm{d}x$, or that have the differential elsewhere than as a factor ? Let me give couple of examples on what I'm thinking of:

$\int 1$

If this is valid notation, I'd expect it to sum infinite ones together, thus to go inifinity.

$\int e^{\mathrm{d}x}$

Again, I'd expect this to go to infinity as $e^0 = 1$, assuming the notation is valid.

$\int (e^{\mathrm{d}x} - 1)$

This I could potentially imagine to have a finite value.

Are any such integrals valid? If so, are there any interesting / enlightening examples of such integrals?

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– Rasmus Aug 16 '10 at 9:42
– Rasmus Aug 16 '10 at 9:46

6 Answers

First of all Non-standard Analysis makes nonsense like $\frac{\mathrm{d}y}{\mathrm{d}x}$ into a meaningful fraction rather than just a "symbol" (whatever that means).

The first and simplest example of differentials being used beyond the $\int f(x)\mathrm{d}x$ format is would be multidimensional integration:

$$\int_V \mathrm{d}x \mathrm{d}y \mathrm{d}z = \int \left(\int \left(\int \mathrm{d}x \right) \mathrm{d}y \right) \mathrm{d}z$$

which gives the volume of the solid $V$.

The next example is Greens Theorem from vector calculus,

$$\int A(x,y) \mathrm{d}x - B(x,y) \mathrm{d}y = \int (\partial_1 B - \partial_2 A) \mathrm{d}x \mathrm{d}y$$

It is also possible to forget integration completely and just use equations with differentials in them to solve calculus problems.

So you can see the standard format is not even close to the whole picture, If you want to integrate terms like $\int (e^{\mathrm{d}x} - 1)$ please do! There is absolutely nothing to stop you figuring out from scratch how to solve this sort of integral and making the theory rigorous.

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Thanks for the answer. I think it's closest to answering my original question. Also, especially thanks for the link to the intro to non-standard analysis. I've been interested in learning about hyper real numbers for a while now, but every article I've looked at so far has been kind of terse. – Sami Aug 13 '10 at 3:00

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You write «If you want to integrate terms like $\int(e^{dx}-1)$ please do!» but you don't mean integrate: you mean find a meaning for the notation. You say «how to solve this sort of integral» but that integral does not make sense according to standard definitions of the integral, so it simply does not make sense to 'solve' it. You can of course come up with a theory that does give sense to such notations, but coming up with such a thing is not computing integrals... – Mariano Suárez-Alvarez♦ Oct 21 '10 at 14:20

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@Mariano Suárez-Alvarez, that is splitting hairs, everything you said is formally correct but there's not much to be learned from it. – anon Oct 21 '10 at 16:34

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@muad: one cannot compute things which do not make any sense under the definitions one uses. That is not splitting hairs! – Mariano Suárez-Alvarez♦ Oct 21 '10 at 16:58

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@muad: I have seen way too many students 'compute' things which do not make any sense with the definitions they have... I would say that recognizing when something is defined or not is one of the most important things one has to learn in order to do any kind of meaningful math. – Mariano Suárez-Alvarez♦ Oct 22 '10 at 16:26
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I think your question here shows that, while you have been using these symbols, you haven't really been given a proper motivation for where they came from.
Let's go back and consider how we came up with the idea of an integral. In a typical class, you will see a lot of pictures like this:

We find the area under the curve by summing up the area of all these little rectangles. If we wanted to write an expression for the area, it would look like:

The Σ means that we are computing a sum. We are adding the areas of the rectangles, which we have numbered 1 through n, to get the complete area under the curve. The area of each rectangle is given by multiplying the height by the width. The height is given by f(xi) because the base of the rectangle is at 0, and the top of the rectangle is where it meets the function f. The Δx represents the width of each rectangle.
When we find the integral, we are taking the limit of this sum as the number of rectangles goes to infinity, and each individual rectangle becomes infinitesimally tiny. You can think of the dx as the equivalent of Δx: it represents the infinitesimally small width of each rectangle that we added up to get the area.

Once you realize this, we can see why integrals only make sense when written ∫f(x)dx. Because we are adding up the areas of rectangles that have height f(x) and width dx. If you try to interpret the expressions you wrote in this way, you will see that they do not really make sense as integrals: you are not summing up rectangles, so you are not finding an area under a curve.

You could, of course, define your own notation in which those expressions behave the way you expect them to, but all mathematical notation is driven based on what people find useful, and what people can agree on and easily understand. Your reuse of the integral sign and dx that people are used to seeing in a particular context will probably result in few people adopting your definition.

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The summation formula is a beautiful analogy. – Justin L. Jul 29 '10 at 5:18
Actually, the motivation for my question was me trying to understand, if there was a lim representation for integration like the one for differentiation. I imagined it to look something like: int A to B { f(x) dx } = lim b -> infinity { SUM a=1 to b { f(A + (B - A) * a / b) * (1 / b) } }. And within that definition, integrals such as int A to B { f(x) } = lim b -> infinity { SUM a=1 to b { f(A + (B - A) * a / b) } } would seem to make sense, too. – Sami Jul 29 '10 at 12:11
@Sami: There is a limit definition for the integral. However, even for the standard Riemann integral, the rigorous definition can appear quite arcane. A less general definition may look something like this. Like I said in my answer, you can adjust the expression as you have in your comment, but the result is not useful in the same way the standard integral is, it only shares a similarity in form. – Larry Wang Aug 3 '10 at 5:22
Justin L, indeed - in the Non-standard analysis it is a beautiful definition! – anon Aug 12 '10 at 20:21

When you write $\int f(x) dx$, the whole of $\int ... dx$ is an indivisible symbol, just as the $d/dx$ is an indivisible symbol when you write $df/dx$.

Of course, there are reasons why the notation is as it is, but trying to manipulate it like you suggest in $\int e^{dx}$, for example, is simply meaningless.

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I could imagine e^{dx} being used as notation for a Riemann-Stieltjes integral, but it's sort of weird notation. – Qiaochu Yuan Jul 29 '10 at 0:06
I'd love to know what the downvote was for! :) – Mariano Suárez-Alvarez♦ Jul 29 '10 at 1:22
We should just write $\int f$ or $\int \lambda x. f(x)$ or something, if the $\mathrm{d}x$ is meaningless and indivisible. – anon Aug 12 '10 at 19:50

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@muad, why should we? Should we also drop the $df/dx$ notation? There is absolutely no reason why notations which are 'split' in this way should not be used. – Mariano Suárez-Alvarez♦ Oct 21 '10 at 14:17

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Well, the immense majority of people disagree with you :) – Mariano Suárez-Alvarez♦ Oct 21 '10 at 16:59
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No, it's not valid. The dx in the integral is a representation of the fact that the integral is obtained as an area, so multiplying the "average" of the function value at each point by an infinitesimal interval.

As the manner in which we don't calculate the area does not change, the notation does not change.

There are different notations that are used when the integral is over a curve, or over more than variable (thus leading for example to volumes).

The d(variable) notation is also used as a reminder that the integral is against a specific variable and not another, e.g. that int x/y dx differs from int x/y dy.

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In the context of calculus, $dx$ simply means 'integrate with respect to $x$'. Some books even omit $dx$ entirely because 'of course we're integrating with respect to $x$'. The $dx$-bit does not get a proper meaning before differential forms.

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The closest thing I've seen is setting $D = \frac{d}{dx}$ and using

$e^D = \displaystyle \sum_{k=0}^\infty {\frac{D^k}{k!}} = 1 + D + \frac{D^2}{2!} + \cdots$

as a new operator on a function. So you can write, for example.

$e^D \left\{ e^x \right\} = e^{x+1}$

Remember in the modern sense we think of $$\int dx$$ as an operation on a function (namely finding the family of primitives), thus we don't split the symbols up. Thus $$\int 1$$ makes no sense.

In other topics, one can think of the $dx$ in a different way than in the elementary task of finding primitives, such as in the Riemann-Stieltjes integration, where you have

$$\int_{I} f d\alpha$$ or $$\int_{a}^b f d\alpha$$

Where you integrate over an interval $I$, and $f$ and $\alpha$ are functions.

Similarily, you usually write $f'' = \frac{d^2}{dx^2}\left\{f\right\}$ since you are applying the operator $\frac{d}{dx}$ two times, which is an abbreviation of $\frac{d}{dx}\left\{\frac{d}{dx}\left\{f\right\}\right\}$

I'm sorry Rasmus but I know nothing about differential forms.

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Huh, the exponential map of the derivative operator... I've never seen that before. So $e^D(ax^2 + bx + c) = 1 + (2ax + b) + a$? Is there any context in which this is useful / interesting / significant? – Rahul Narain Feb 1 '12 at 5:23
In Operational Calculus. I think I saw this on Spiegel's Applied Differential Equations, but now I'm not sure. – Peter Tamaroff Feb 1 '12 at 5:42
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