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Infinitesimals as interval limits in integration

  1. Apr 8, 2015 #1
    Ok so what I want to know is, is this valid? If so what does it mean?

  2. jcsd
  3. Apr 8, 2015 #2


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    The notation is unusual. It might simply mean F(0)dx.
  4. Apr 8, 2015 #3
    The integral comes from finding the work done by a force over a distance dx. The force may or may not be variable so I needed to prove the variableness of F did not matter over an infinitesimal for the purposes of determining the amount of work done.

    So dw = F⋅dx => F0⋅dx

    So integrating over the distance dx would prove this.


    I'm doing a proof that I'd like to make rigorous. This makes sense to me visually - we are simply looking at the area of the first strip of dx - which would be the initial value of F. However I cannot find a way to prove it mathematically.
    Last edited: Apr 8, 2015
  5. Apr 9, 2015 #4


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    You first need to define an integral with an upper limit dx.
  6. Apr 10, 2015 #5
    An upper limit of dx would be the area of the first strip of dx. I need to prove the above.
  7. Apr 10, 2015 #6
    Doh. The area of that first strip would be the height (F0) by the width (dx.) Ok problem solved.
    Last edited: Apr 10, 2015
  8. Apr 10, 2015 #7


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    Notation quibble: dx is used for the differential. If you are trying to describe a small non-zero width, use Δx.
  9. Apr 11, 2015 #8
    It has to be along a distance dx.
  10. Apr 11, 2015 #9


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    Like mathman said, use Δx. dx has its own meaning. Δx can represent some small distance along the x-axis.
  11. Apr 12, 2015 #10
    dx is the width of each strip. dx would then be the width of the first strip. Δx is the width of a finite number of strips, which is not what I am looking for.
  12. Apr 12, 2015 #11


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    Thirding what the others said: the width of the first strip is ##\Delta x_0##, the width of the second strip is ##\Delta x_1## etc. The width of a finite number of strips is ##\Sigma \Delta x_n##.

    dx is not used for the width of a strip.
    Last edited: Apr 12, 2015
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