Another really basic question... this time regarding integration.

  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. pwsnafu

    pwsnafu 902
    Science Advisor

    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. 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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