Another really basic question this time regarding integration.

  1. Jun 13, 2012 #1
    Given a function f define a new function Sf(x) by summing up all values of f(hj)
    where 0 ≤ jh < x. That is, if k is such that kh is the largest below x, then
    Sf(x) = h[ f(0) + f(h) + f(2h) + .... + f(kh) ]
    We call Sf also the ”integral” or ”antiderivative” of f.

    The teacher who wrote the lecture notes I'm reading through gives an example of integration. He evaluates Sf(x) for f(x)=1. I don't understand the first sentence:

    We have Sf(x) = 0 for x ≤ h.

    Why? Sorry for being such a n00b, but I don't understand. Please help me.

    Thanks,
    Mathguy

    By the way, he verifies that the js in the definition are integers.
     
  2. jcsd
  3. Jun 13, 2012 #2

    pwsnafu

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    Err, the integral is what you get if you take the limit h -> 0.
    Is that word-for-word what is written there?

    That doesn't seem right. jh is allowed to equal 0, so the largest integer k such that
    0 ≤ kh < x ≤ h is when k=0. So Sf(x) = h f(0) = h.

    Edit: Maybe he means x < 0?
     
    Last edited: Jun 13, 2012
  4. Jun 14, 2012 #3
    Well, Yes, that is word-for-word, but I think he's doing a "preliminary" definition before the real definition. And I was thinking the same thing, because Sf(x) isn't defined for x<0.
     
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