# Understanding dx in Integration

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1. Jan 29, 2016

Hello, I am currently in my first year of college, and I already took calculus in high school. I was able to solve all the problems, but I feel like I didn't understand everything conceptually.
When integrating dy/dx=x you get,
∫x dx=1/2x2.
But what exactly happened to the dx, why did it disappear? I thought that dx was an infinitesimally small number, representing the width of the rectangles in a Riemann Sum.
Also, I tried solving this question geometrically, but I think that I might be doing something wrong. Starting from any point x, and using left endpoint rectangles, I got that the heights of the rectangles are x,x+dx,x+2dx,... And the width is just dx. So the area under the curve y=x is dx(x+(x+dx)+(x+2dx)+(x+3dx)...) which is equal to:
dx(x+x+x...+dx(1+2+3+4...))? Am I misinterpreting dx or can I simplify this sum even further to get 1/2x2? Thank you.

2. Jan 29, 2016

### andrewkirk

Yes you can simplify it further.
Say the integral is from 0 to 1. Divide the length up into n equal segments so that $dx=\tfrac{1}{n}$.
Then what you have written as dx(x+x+x...+dx(1+2+3+4...)) is actually (substituting 0 for x)

$$dx\sum_{k=1}^n\left(0+k\,dx\right) =\tfrac{1}{n}\left(\sum_{k=1}^n\tfrac{1}{n}k\right) =\tfrac{1}{n^2}\left(\frac{n(n+1)}{2}\right)=\frac{1}{2}+\frac{1}{2n^2}$$

which tends towards $\tfrac{1}{2}$ as $n\to\infty$.

3. Jan 30, 2016

Oh ok, thank you. I was able to use the summations to also get the integral from 0 to any value b. By replacing dx with b/n I got the same Riemann sums as what I learned in high school.

4. Jan 30, 2016

### Staff: Mentor

That's fine, but the integral you showed, ∫x dx=1/2x2, is an indefinite integral, so does not evaluate to a number.

The principal use of dx in definite or indefinite integrals is to indicate what the variable of integration is. For example, in $\int my^2 dy$, the dy tells us that the variable is y, and that for the purposes of integration, m is a constant.

Also, with indefinite integrals, be careful to include the arbitrary constant. In your example, it should be $\int x~dx = \frac 1 2 x^2 + C$.

5. Feb 2, 2016

### zinq

"But what exactly happened to the dx, why did it disappear? I thought that dx was an infinitesimally small number, representing the width of the rectangles in a Riemann Sum."

For an indefinite integral, which is what you have displayed with ∫ x dx, think of this as just indicating you want the antiderivative(s) of the integrand. Since the derivative of x2/2 is in fact the integrand x, x2/2 is one antiderivative. If you slide the graph of y = x2/2 by any amount vertically (i.e., adding a constant to x2/2), this will not change the slope of the function; i.e., its derivative will still be x for any constant you add. Thus

∫ x dx = x2/2 + C

is your most general family of antiderivatives of x.

Do not think of the dx as an infinitesimally small number, but rather as an instruction in the definition of an indefinite (or definite) integral as to how to intepret the rest of the notation.

Last edited by a moderator: Feb 3, 2016
6. Feb 3, 2016

### Staff: Mentor

I'm glad you agree with me. See post #4.

7. Feb 3, 2016

### mfig

I disagree with this slightly. The dx is not merely an instruction or indication, it is part of the differential itself. When one sees this:

$\int x^3~dx$

It can be understood as, "find the function whose differential is $dy = x^3~dx$." So, what function has that differential? Well I think it is easy to see that if we set $y = \frac{x^4}{4}$, then the differential of y is $dy = x^3~dx$.

8. Feb 3, 2016

### Staff: Mentor

It's kind of "six of one, a half-dozen of the other." In other words, you can look at $\int x^3~dx$ as a) finding that antiderivative of $x^3$, or b) finding the "antidifferential" (not sure if that's a word) of $x^3 dx$. Either way, dx doesn't play much of a role, other than to indicate with respect to which variable integration/antidifferentiation should be done.

9. Feb 3, 2016

### mfig

Yeah, maybe. I am just uncomfortable with any response that seems to indicate it has no real meaning other than as a pointer. After taking differential forms, I saw that these things have all kinds of uses and even an algebra associated with them. To say that the differential of $y = x^2$ is given by $dy = 2xdx$ , is to say something with definite meaning. We cannot dismiss, in general, the dy and dx as mere notation.

On the other hand, for someone who is just learning integration, perhaps treating them as pointers is just fine.

Last edited: Feb 3, 2016
10. Feb 3, 2016

### Staff: Mentor

I meant that only in the context of integrals, not in general.
I agree that they have importance other than as indicators of what the variable of integration is. Many students don't bother to write the dx (or appropriate differential) in an integral, but come to grief in integration techniques such as trig substitution or integration by parts.

11. Feb 3, 2016

### bcrowell

Staff Emeritus
See http://math.stackexchange.com/questions/200393/what-is-dx-in-integration .

The notion that dx is only a pointer or punctuation is misguided revisionist history. The dx represents an infinitesimal number. It doesn't matter whether you're talking about a definite integral or an indefinite integral. An indefinite integral can be thought of as a definite integral in which one of the bounds of integration is the independent variable.

12. Feb 5, 2016

### zinq

bcrowell: The history of mathematics is one thing; modern mathematics is another thing. As concepts become clarified, definitions become more precise. What dx once was is not the same thing as what dx is now.

In advanced mathematics beyond elementary calculus (and which is sometimes taught in honors calculus), one learns that dx can be thought of fully rigorously as a linear function on tangent vectors. As spamanon mentions above.

There are also other ways of interpreting it that are closer to the concept of an infinitesimal, but in a modern rigorous way, far removed from the vague infinitesimals of Newton and Leibniz. Usually one must take a rigorous course in mathematical logic to fully understand these interpretations.

13. Feb 6, 2016

### lavinia

As zinq has said, in modern integration theory the things that get integrated are differential forms. Differential forms are multilinear alternating tensors.

d$x$ is a tensor of degree 1. In general, a 1 form can be expressed with respect to a basis for the cotangent space as is a sum, $Σf_{i}$d$x_{i}$ .

14. Feb 6, 2016

### bcrowell

Staff Emeritus
Your first paragraph gets the history wrong. Infinitesimals have persisted since Newton and Leibniz without interruption; scientists and engineers have been using a consistent set of practices ever since that time.

Your second paragraph gets the math wrong. Yes, one can choose to interpret a dx in certain contexts using the language of differential forms. No, that is not the only way to interpret it, and in fact it doesn't work well in many cases (e.g., second derivatives).

Your third paragraph is also wrong. No, it is not necessary to take "a rigorous course in mathematical logic to fully understand these interpretations." For a counterexample, see the freshman calculus book by Keisler (free online): http://www.math.wisc.edu/~keisler/calc.html . Your third paragraph also contradicts your first and second paragraphs, which is odd -- the first and second are written as if you were ignorant of NSA, while the third shows that you know of it.

This is wrong for the same reasons.

Last edited: Feb 6, 2016
15. Feb 6, 2016

### PeroK

Suppose, for the sake of argument, I decided to write $Int[f(x), x]$ instead of $\int f(x) dx$. Is there a theorem or result that would not hold or could not be proved using my notation, but could be proved using the standard form of the integral?

16. Feb 6, 2016

### bcrowell

Staff Emeritus
The standard notation and the notation you suggest are isomorphic to each other, i.e., you can translate back and forth without ambiguity. Therefore the answer to your question is no.

However, that doesn't mean that they're both equally good notations. The standard (Leibniz) notation is good for a variety of reasons, one of the most important ones being that it makes sense of the units. For example, if I integrate velocity with respect to time, I get position. Notating it in Leibniz notation, $x=\int v dt$, we can see that the units do make sense: meters=(meters/second)(seconds).

17. Feb 6, 2016

### zinq

Thanks for the link to that infinitesimal calculus book, bcrowell. As far as I can tell, however, the book merely asserts that infinitesimals exist and goes from there. Which to my way of thinking doesn't create understanding.

How would you define an infinitesimal?

18. Feb 6, 2016

### bcrowell

Staff Emeritus
The quantity x is a positive infinitesimal if $x>0$, but $x<1$, $x<1/2$, $x<1/3$, ... (The ... means that it has to satisfy all inequalities of this form.)

The real numbers have the Archimedean property, which says that they do not include any infinitesimals. The Archimedean property is logically implied by the completeness property of the reals. The reals are the unique complete, ordered field.

Last edited: Feb 6, 2016
19. Feb 6, 2016

### zinq

You have given a defining property of a positive infinitesimal, and then stated that the real numbers do not include any infinitesimals.

So, where do we find them? What is an example of one?

20. Feb 6, 2016

### PeroK

Yes, I agree it's good notation. But, it is "only" notation! But, then, what isn't just notation when it comes down to it?

21. Feb 6, 2016

### PeroK

Which, unfortunately, rather suggests that the Riemann integral can't be defined in terms of standard Real Analysis.

22. Feb 6, 2016

### zinq

The Riemann integral most assuredly can be, and is, defined in terms of standard real numbers.

23. Feb 6, 2016

### PeroK

Not if it's got a "non-real" differential in it! But, if that differential is merely notation ...

24. Feb 6, 2016

### bcrowell

Staff Emeritus
You seem to be making some assumptions about what a number is, how it relates to the real world, and the role of the real number system. If you examine these assumptions carefully, they don't hold.

The real numbers are not "really" "real" in the sense of relating perfectly to the "real" world. For example, the real number system has a distinction between rational and irrational numbers, but such a distinction can never be empirically verified for any real-world measurement. For example, there is no way, even in principle, to determine whether the mass of a hydrogen atom is a rational number when expressed in units of kilograms.

So when you ask "where do we find them," the answer is that we don't, and in fact this applies even to the real number system, and, arguably, the integers. (If you enthusiastically assert this philosophical position for the integers, you are what's known as an ultrafinitist.)

You ask "what is an example of one?" It is possible to make up number systems in which there are concrete examples of infinitesimals. A pretty rich system of this type is the Levi-Civita numbers. Here is a calculator I wrote that allows you to play with the Levi-Civita numbers: http://lightandmatter.com/calc/inf/ . However, when you do calculus it is never necessary or desirable to define a specific, concrete infinitesimal. This is really not so different from the real numbers. The vast majority of real numbers can never be defined, because there are uncountably many real numbers, but only countably many definitions.

25. Feb 6, 2016

### zinq

What "assumptions" am I making? All I did was ask questions.