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Integration Summation Notation

  1. May 4, 2010 #1
    Okay I've seen how crazy Riemann sums can get in real analysis and I've noticed a heirarchy of notation.

    The Stewart/Thomas etc... kinds of books use;

    [tex]\lim_{x \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x[/tex]

    Where;

    [tex]\Delta x = \frac{b - a}{n} and x_i = a + i\Delta x[/tex]

    Then the books like Apostol and Bartle's real analysis use;

    [tex]\lim_{x \to \infty} \sum_{i=1}^{n} f(x_i) (x_i - x_i_-_1)[/tex]


    and what I'd like to know is how to calculate the (x_i - x_i_-_1) for some equation like;

    f(x) = x² integrated from 2 to 8. I can do the Δx = (b - a)/n version fine but how do you work the newer notation?

    in f(x_i) (x_i - x_i_-_1) I would assume f(x_i) would use any endpoint, i.e. the right endpoint being a + iΔx but how do you make sense of the (x_i - x_i_-_1)???
     
  2. jcsd
  3. May 5, 2010 #2
    As you say, they both essentially mean the same thing. To see this, just substitute [tex](i.\frac{b - a}{n}[/tex] for [tex]x_i[/tex] and [tex](i-1).\frac{b - a}{n}[/tex] for [tex]x_{i-1}[/tex] in the latter notation, and simplify.

    [tex]x_i[/tex] just means the "value you get after adding [tex]i\frac{b-a}{n}[/tex] to [tex]a[/tex] ".

    I'm not sure if this answers your question.
     
    Last edited: May 6, 2010
  4. May 6, 2010 #3
    Thomas is in BIG TROUBLE if he actually says [itex]x \to \infty[/itex] ... but I think it is just sponsoredwalk who is mistaken ...
     
  5. May 6, 2010 #4
    That answered my question perfectly thanks, I was a bit confused because the i was in the subscript.

    Yeah that's actually [tex] \lim_{n \to \infty} [/tex] :redface:
     
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