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Graphic Interpretation for Æ©f(x)Îx

  1. Dec 21, 2013 #1
    Graphic Interpretation for Ʃf(x)Δx

    Hello!

    I was trying to understand what means:
    [tex]\sum_{x_0}^{x_1}f(x)\Delta x[/tex]
    (when Δx = 1 and x ∈ Z, ie, a "discrete integration", topic very comun in discrete calculus).

    I computed the result so:
    [tex]\sum_{1}^{4}x^2\Delta x=\left [\frac{1}{3}x^3-\frac{1}{2}x^2+\frac{1}{6}x \right ]_{1}^{4}=F(4)-F(1)=14[/tex]
    and I sketched the graphic:
    imagem.jpg

    However, the Maple computes the result as:
    [tex]\sum_{1}^{4}x^2\Delta x=>\sum_{1}^{4}x^2\cdot 1=>\sum_{1}^{4}x^2=30[/tex]
    Given a different result of calculated for me. Why?
     
  2. jcsd
  3. Dec 21, 2013 #2

    AlephZero

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    Maple is calculating the sum following the conventional rules of notation for ##\sum_1^4##, i.e. the sum is 1 + 4 + 9 + 16 = 30.

    In discrete calculus you are only summing 3 items, not 4. That is why you got 1 + 4 + 9 = 14.

    Maybe there is an option in Maple to use discrete calculus notation, but I don't know about that.
     
  4. Dec 24, 2013 #3

    HallsofIvy

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    This is not a very good notation! It says you are summing from [itex]x_0[/itex] to [itex]x_1[/itex] but does NOT say what step you are using- what [itex]\Delt x[/itex is.

    You appear to be assuming that [itex]x_0= 1[/itex] and [itex]x_1= 4[/itex[ are the only values used- that is, that [itex]\Delta x= 4- 1= 3. But this is a sum, not a difference. It is [itex](1)^2(3)+ (2)^2+ 3= 3+ 12= 15[/itex].

    That graphic shows you using [itex]\Delta x= 1[/itex], so that x takes on values of 1, 2, and 3:
    [itex]1^2(1)+ 2^2(1)+ 3^2(1)= 1+ 4+ 9= 14[/itex]

    However, the Maple computes the result as:
    [tex]\sum_{1}^{4}x^2\Delta x=>\sum_{1}^{4}x^2\cdot 1=>\sum_{1}^{4}x^2=30[/tex]
    Given a different result of calculated for me. Why?[/QUOTE]
    That sum would NOT approximate the integral from 1 to 4 because it includes a "rectangle" between x= 4 and x= 5: 1+ 4+ 9+ 16= 30.
     
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