Why do we write dx in indefinite integrals

In summary, the Riemann integral notation uses dx (or dy) as a reminder that the integral is approximated by Riemann sums and to indicate which variable is being integrated over. In the indefinite integral, the dx (or dy) is used in differential notation to represent an infinitesimal change in the function being integrated. This notation is also useful in finding antiderivatives using the chain rule.
  • #1
KingCrimson
43
1
in understand why we write the dx in riemann integral , but in the indefinite integral why do we use that ?
what is the relation between the area under a curve , and the antiderivative of that of that curve ??
 
Physics news on Phys.org
  • #2
Because it is a very tempting "abuse" of notation?
 
  • #3
Let [itex]F[/itex] be an antiderivative of a function [itex]f[/itex] such that [itex]F(0)=0[/itex] , you can see that [itex]F(x)[/itex] is the area under the graph of [itex]f[/itex] from [itex]0[/itex] to [itex]x[/itex] thanks to the beautiful Fundamental Theorem of Calculus:
[tex]F(x)=\int_0^x f(s)ds.[/tex]

Here you see the power of this theorem, because it connects two different things, which are the antiderivative and the area under the graph.
 
Last edited:
  • Like
Likes 1 person
  • #4
ndjokovic said:
Let [itex]F[/itex] be an antiderivative of a function [itex]f[/itex] such that [itex]F(0)=0[/itex] , you can see that [itex]F(x)[/itex] is the area under the graph of [itex]f[/itex] from [itex]0[/itex] to [itex]x[/itex] thanks to the beautiful Fundamental Theorem of Calculus:
[tex]F(x)=\int_0^x f(s)ds.[/tex]

Here you see the power of this theorem, because it connects two different things, which are the antiderivative and the area under the graph.

aha my apologies as i have not been introduced to this concept .
thanks for everything
 
  • #5
The other answer is "so we know what variable we're integrating with respect to".
 
  • #6
pasmith said:
The other answer is "so we know what variable we're integrating with respect to".

Also, when we integrate something, what are really finding? The answer is the function to find area. So what is area? Length*Width. The length is the function to integrate and the width is dx. So, we have length*width for area as well.
 
  • #7
An integral is an infinite sum of terms, and an infinite sum of finite terms is infinite. We include the differential to indicate that the terms are infinitesimal. If the terms are infinitesimal, the integral is finite, and thus meaningful. At least that's what my professors say.
 
  • #8
MexChemE said:
An integral is an infinite sum of terms, and an infinite sum of finite terms is infinite.
An infinite sum of finite terms is not necessarily infinite. For example, this infinite series converges to a finite number:
$$ \sum_{n = 1}^{\infty}\frac 1 {2^n}$$
MexChemE said:
We include the differential to indicate that the terms are infinitesimal. If the terms are infinitesimal, the integral is finite, and thus meaningful. At least that's what my professors say.
I think you are misinterpreting what they're saying.
 
  • #9
it helps us to guess the antiderivative using the chain rule, i.e. the integrand 2x.sin(x^2)dx becomes sin(x^2)d(x^2) = -dcos(x^2)
 
  • #10
Consider the indefinite integral, which is just antidifferentiation. For the indefinite integral of f(x) we want to find the function y = F(x) such that F'(x) = f(x). In differential notation that is dy/dx = f(x) or else dy = f(x) dx. So antidifferentiation is just finding the differential of F(x) rather than the derivative (dy vs dy/dx).

The other reasons as mentioned is with respect to the definite integral it reminds us that we are summing areas over small increments of x - that is Δx. And recall the definition of the differential as an approximation of an increment around some x provided f(x) is differential around that x. So

dy = f'(x) Δx ≈ Δy

As Δx approaches zero dy is a better and better approximation to Δy.

As another note, I believe the first intuition that led to the fundamental theorem of calculus goes something like this: the rate of change of the area under a curve is equal to the height of the curve at that point. Not sure why I don't see this intuition stated often enough. If the curve is higher at some point x then it stands to reason the rate of change of the total area under the curve is changing faster or slower accordingly.
 
  • #11
I was always taught that dx is delta x; the width of each of the rectangles in the area.
 
  • #12
Kant Destroyer said:
I was always taught that dx is delta x; the width of each of the rectangles in the area.

I do admit that I have been confused by the fuzzy use of differential notation and I still am quite often. But I'm fairly certain that dx is an approximation of Δx. The increment is the actual change in a value whereas dx is an approximation of that change using the derivative. It is certainly possible that your teacher was confused as well.

This is the book I have used to learn and where I obtained my understanding of the differential. This book is fantastic. I couldn't recommend it enough and it is 3$ used. Mine is old, old.

https://www.amazon.com/dp/0060441070/?tag=pfamazon01-20
 
Last edited by a moderator:
  • #13
The dx (or dy or whatever) in a Riemann integral doesn't mean anything on its own. The notation is meant to be a reminder of the fact that the integral is approximated by Riemann sums.
$$\int_a^b f(x)\mathrm dx \approx \sum_{k=1}^n f(x_k)\Delta x_k.$$ And as pasmith said, it's also a reminder of which one of the variables we're integrating over. For example,
$$\int_a^b yx^2 \mathrm dx=y\left(\frac{b^3}{3}-\frac{a^3}{3}\right),$$ but
$$\int_a^b yx^2 \mathrm dy=x^2\left(\frac{b^2}{2}-\frac{a^2}{2}\right).$$ Here the dx or dy is letting us know if the function that should be integrated from a to b is the 2nd degree polynomial function ##x\mapsto yx^2## or the 1st degree polynomial function ##y\mapsto yx^2##.
 

FAQ: Why do we write dx in indefinite integrals

1. Why do we write dx in indefinite integrals?

The "dx" in an indefinite integral represents the variable of integration. It is a notation used to indicate that the integral is with respect to that variable. It is necessary to include the "dx" to specify which variable is being integrated.

2. Can we use any other letter instead of "dx" in indefinite integrals?

Yes, we can use any letter as long as it is consistent with the variable of integration used in the integral. For example, if we are integrating with respect to "x", we can use "dx", "dy", or "dz" as the "dx" notation.

3. Is the "dx" in indefinite integrals just a placeholder?

No, the "dx" is not just a placeholder. It has a specific mathematical meaning as the variable of integration. It is important to include the "dx" to indicate the variable of integration and to make the integral well-defined.

4. Why is the "dx" written after the integrand in indefinite integrals?

The "dx" is written after the integrand to indicate that the integral is with respect to the variable "x". This notation follows the conventions of mathematical notation and helps to clearly specify the variable of integration.

5. Is it necessary to include the "dx" in indefinite integrals?

Yes, it is necessary to include the "dx" to indicate the variable of integration and to make the integral well-defined. Omitting the "dx" can lead to ambiguity and incorrect solutions.

Similar threads

Replies
4
Views
2K
Replies
9
Views
2K
Replies
20
Views
3K
Replies
11
Views
34K
Replies
11
Views
2K
Replies
6
Views
2K
Replies
3
Views
2K
Back
Top