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Infinitesimals in integration vs delta x in summations

  1. Oct 18, 2012 #1
    I first had a question regarding infinitesimals. What does it mean when the infinitesimal is at the beginning of the integral? For example:

    is this the same as
    ∫f(x)dx ?

    My second question was how to convert a summation to an integral and a summation into an integral. Thanks a lot,

  2. jcsd
  3. Oct 19, 2012 #2


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    Yes, those are just two different notations for the same thing.

    In a physics class (where mathematics is done without rigor), you just do something like this:
    $$\sum_i F(x_i) =\sum_i \frac{F(x_i)}{\Delta x_i}\Delta x_i =\sum_i f(x_i)\Delta x_i \approx\int f(x) dx,$$ where ##f(x_i)=F(x_i)/\Delta x_i##.

    If you want a rigorous answer, I think you will have to make the question more specific.
  4. Oct 19, 2012 #3


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    Science Advisor

    This may be more picky than necessary, but you are actually asking about the differential, dx. An "infinitesmal" is a completely different matter.
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