Question on integral notation (dt, dx, etc.)

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    Dx Integral Notation
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Discussion Overview

The discussion revolves around the proper use of integral notation, particularly concerning the variable of integration and the limits of integration. Participants explore whether it is acceptable to use the same variable in both the limits and the differential notation, as well as the implications of such usage in mathematical expressions.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions if using a differential notation (dt) that matches the variable in the limit (t) is acceptable, suggesting that it might lead to confusion.
  • Another participant argues that if the integral is expressed as ∫ f(t) dt, it must have limits that are also in terms of t, indicating that t cannot depend on itself.
  • A different viewpoint suggests that if t represents a specific value within a range (1
  • One participant clarifies that using the same variable for different purposes within the integral notation is considered an abuse of notation, emphasizing that the differential (dt) is a dummy variable and can be replaced with any symbol.
  • Examples are provided to illustrate the confusion that arises when the same variable is used in both the limits and the integrand, with a specific example showing how using a different variable (tau) can clarify the expression.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of using the same variable in both the limits and the differential notation. There is no consensus on whether this practice is acceptable, and the discussion remains unresolved.

Contextual Notes

Participants highlight the potential for confusion when the same variable is used in multiple contexts within integral notation. The discussion also touches on the importance of clarity in mathematical expressions, but specific limitations or assumptions are not fully explored.

hulgy
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Say the limit of the integral is from 0 to t and the integral is ended with a dt. Is this okay?

Generally, all the integrals I see with a variable limit end with a d-letter that is not the same as the variable in the limit. ie: limit is from 0 to t, ends with du
 
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Welcome to PF!

Hi hulgy! Welcome to PF! :smile:
hulgy said:
Say the limit of the integral is from 0 to t and the integral is ended with a dt. Is this okay?

Noooo …

If you have an ∫ f(t) dt, that has to be between limits of t …

t can't depend on itself. :wink:​

(once you've integrated over a variable like t, that variable should disappear, not come back to haunt you!)
 


tiny-tim said:
Hi hulgy! Welcome to PF! :smile:


Noooo …

If you have an ∫ f(t) dt, that has to be between limits of t …

t can't depend on itself. :wink:​

(once you've integrated over a variable like t, that variable should disappear, not come back to haunt you!)

Hmmm... what if t stood for a value like 1<t<3 which you would have to place into the 0 to t limit (this would technically eliminate the variable...right?). Would the integral as a whole still be wrong then?
 
hulgy said:
Hmmm... what if t stood for a value like 1<t<3 which you would have to place into the 0 to t limit (this would technically eliminate the variable...right?). Would the integral as a whole still be wrong then?

Not following you :confused: …

can you write the integral out in full? :smile:

(you can write integrals like this: [noparse]∫ab[/noparse] … ∫ab :wink:)
 
tiny-tim said:
Not following you :confused: …

can you write the integral out in full? :smile:

(you can write integrals like this: [noparse]∫ab[/noparse] … ∫ab :wink:)


∫0t (50)dt for 1<t<3
 
hulgy said:
∫0t (50)dt for 1<t<3

hmm …

If you mean ∫0s (50)dt for 1<s<3, then that's fine.

But no, you can't use t for two different things.
 
That is a bit of abuse of notation -- not that you won't see such abuse.

The dt inside the integral is a dummy variable. It might as well be tau, or x, or anything. It has nothing to do with the upper limit of the integration.

This is at best confusing:

f(t) = \int_0^t \cos t\, dt

What is df/dt?

If the above integral is expressed as

f(t) = \int_0^t \cos \tau\,d\tau

then it is pretty clear that f(t)=\sin t and that df/dt = \cos t.


Occasionally you will run across things like

f(t) = \int_0^t \cos (t-\tau)\,d\tau

The t inside the above integral is not a dummy variable. As far as the integration is concerned that t inside the integral is a constant.
 
I guess I'm going to get points off for notation on my AP calc test then. Well thanks for all the help, really cleared things up.
 

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