What is the Integral of x_2-x_1?

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Discussion Overview

The discussion revolves around the interpretation and application of integrals, particularly in the context of changes in physical quantities, such as velocity and height, as related to raindrop dynamics. Participants explore the implications of finite versus infinite integrals and their relationship to the concept of 'delta' or change.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant references a partial differential equation from a lecture on interrill erosion processes, suggesting a connection to the discussion topic.
  • Another participant questions the clarity of the physical process represented by 'Δ' and critiques the transition from Δx = ∫Vx.dt to Δx = ∫ΔVx.dt as lacking physical sense.
  • A third participant expresses that they resolved their confusion by recognizing that a finite integral inherently represents a 'delta' or change, thus negating the need to find the 'integral of a delta'.
  • This same participant contrasts finite integrals with infinite integrals, suggesting that the latter may involve integrating deltas, and provides a graphical interpretation of finite integrals representing areas between curves.
  • A fourth participant shares a link to an external explanation, indicating a potential solution or clarification regarding the integral of delta.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of integrals and the concept of 'delta'. While one participant feels they have resolved their confusion, others raise questions and critiques, indicating that the discussion remains unresolved with multiple competing interpretations.

Contextual Notes

There are limitations in the assumptions made regarding the relationship between changes in height, velocity, and time, as well as the definitions of finite and infinite integrals. These aspects remain open to interpretation and clarification.

Kuhan
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https://docs.google.com/file/d/0Bx0Sv7WqdghaOGR1YU9QUW5icWM/edit

Note: the partial differential equation (on the third line) is quoted from Erpul, G., Gabriels, D., Norton, L.D., 2003. The Combined Effect of Wind and Rain on Interrill Erosion Processes. Lecture given at the College on Soil Physics Trieste, 3 -21 March 2003 (LNS0418015):174-182. http://users.ictp.it/~pub_off/lectures/lns018/15Erpul.pdf [8 August 2012] and he makes a reference to "(Pedersen and Hasholt, 1995)" concerning the equation
 
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You need to be clear in your mind about the physical process intended by the 'Δ'.
You expression for ΔVx is fine, if it means the change in horizontal velocity of a raindrop as it descends from z1 to z2. But then you introduce the notion of a delta in time as though it were independent. For a given raindrop, a delta in vertical height will be accompanied by a delta in time.
To arrive at your final equation, you seem to have shifted from Δx = ∫Vx.dt to Δx = ∫ΔVx.dt, which doesn't make any physical sense to me.
 
Thanks! I managed to solve the problem.
I think the point I got wrong is that when we have a FINITE integral, it already counts as a 'delta' or 'change', thus we don't need to find the 'integral of a delta'

On the other hand, when we have an INFINITE integral, then we may 'integrate deltas'. Then, Δx=∫ΔV.dt if we use infinite integrals.

A clear example would be using graphs. The finite integral already shows us the area BETWEEN two curves or lines, so we don't have to find a 'delta' because the finite integral already gives the 'delta'.
Using a delta in an infinite integral basically makes the infinite integral behave like a finite integral without a delta.
 
I've written an explanation (the answer) here : http://eraserboxtips.blogspot.com/2012/09/the-integral-of-delta.html
 
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