Why is integration with dx in the denominator undefined?

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Discussion Overview

The discussion revolves around the question of why integration with dx in the denominator is considered undefined. Participants explore the implications of this notation and its mathematical validity, touching on concepts of integration and differentiation.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants argue that integration with dx in the denominator is undefined because there is no mathematical need to define it, as stated by one participant.
  • Others express confusion and seek a clearer explanation for why it is considered undefined, questioning the reasoning behind the lack of definition.
  • One participant suggests that dx serves as a symbol indicating the variable of integration rather than representing a multiplication that could lead to division.
  • Another participant compares the question to asking why certain mathematical expressions, like ##{}^25##, are undefined, implying that the lack of definition stems from a lack of necessity.
  • There is a mention of the relationship between integrals and areas, where width and length are involved, but this does not directly address the issue of dx in the denominator.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the reasons for the undefined nature of integration with dx in the denominator, with multiple competing views and ongoing confusion about the topic.

Contextual Notes

Some participants express a need for a more concrete reason for the undefined status, indicating that the discussion lacks a definitive explanation or resolution of the underlying assumptions regarding the notation.

Aows
Hello,
In your opinion, why the integration with dx in the denominator is undefined ?? (as in the attached picture)
xqi876U.png
 
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Aows said:
Hello,
In your opinion, why the integration with dx in the denominator is undefined ??
Could you explain what you mean? What do you think does dx have to do in the integral?
This question is a bit like asking why ##{}^25## is undefined.
 
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Ok, here is the question in clearer way (as in the attached pic)
xqi876U.png
 
This is undefined, because nobody defined it. And nobody defined it, because there is no need to define it. At least no mathematical need. One could attach any meaning to it, but this would cause confusions with the integral ##\int \frac{1}{x}\,dx## which is defined.
 
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I can't understand your answer.
i need a reason why it is undefined.
 
Aows said:
I can't understand your answer.
i need a reason why it is undefined.
What reason do you need and why? A definition is an action. In order to act, people need a motivation. But there is none. It is that simple. Why isn't ##\int x\,\clubsuit \,x## defined? I try to figure out, whether you had something in mind in order to want to define it, or you just picked some symbols and modeled a question with it.

The point is, that ##\int f(x)dx## isn't really a "multiplication". ##dx## is rather a symbol to indicate the variable of integration. Because it is no multiplication, why should there be a division? Of course it is related to a multiplication as we use integrals to calculate areas, which are width times length. So width divide by length is the slope of something, which leads to differentiations, where ##\frac{d}{dx}## is denoted as a quotient. But this isn't a true division either, only an encoded text which describes what really has to be done. So a notation ## "\colon" \, dx## leads to differentiation which is the correspondence to the integration ##"\cdot" \, dx##
 
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Thread closed. As @fresh_42 already said, writing an integral with a differential in a denominator is not defined.
 

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