Usage of dx in Integrals

Question

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?

Answer 1

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.

Answer 2

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(x_i) 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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.