Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?

    eqn1148.png
     
  2. jcsd
  3. Apr 8, 2015 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    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.

    gif.latex.gif

    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

    mathman

    User Avatar
    Science Advisor
    Gold Member

    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

    mathman

    User Avatar
    Science Advisor
    Gold Member

    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

    Mark44

    Staff: Mentor

    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

    pwsnafu

    User Avatar
    Science Advisor

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Infinitesimals as interval limits in integration
  1. Limit in the interval (Replies: 5)

Loading...