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Integral of de absolute value of a derivative

  1. Aug 23, 2011 #1
    Hi!

    1. The problem statement, all variables and given/known data

    I'm trying to work out this integral, without success:

    [itex]\int_{-a}^{a} (1-\left|x\right|/a) \frac{d^2}{dx^2}(1-\left|x\right|/a)[/itex]

    2. The attempt at a solution

    I tried solving this by parts, but i'm stuck:

    [itex]\int_{-a}^{a} (1-\left|x\right|/a) \frac{d^2}{dx^2}(1-\left|x\right|/a)[/itex]
    [itex]=(1-\left|x\right|/a) \frac{d}{dx}(1-\left|x\right|/a)|_{-a}^{a}- \int_{-a}^{a} (\frac{d}{dx}(1-\left|x\right|/a))^2[/itex]


    Now, the first term seems to be wrong; the derivative of abs(x) is not defined, and with the second term, more of that. I tried to split the integral in two parts, for positive and negative x's, but that gives me 0. I think that result is wrong, because the function [itex](1-\left|x\right|/a) [/itex] is concave, and I'm expecting a positive second derivative.

    Well, any input would be highly appreciated.

    Thanks.
     
  2. jcsd
  3. Aug 23, 2011 #2

    Ray Vickson

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    (d/dx)[1-|x|/a] is +1/a for x < 0 and is -1/a for x > 0 (and is undefined at x = 0). If you feel confident applying integration by parts in the presence of such discontinuities, you can use the above to complete the calculation. Alternatively, you can regard |x|/a as the limit of sqrt(e+x^2/a^2) as e --> 0 from above, do the integral for finite e > 0, then take the limit in the final result.

    RGV
     
  4. Aug 25, 2011 #3
    Thanks a lot!
     
  5. Aug 25, 2011 #4

    rude man

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    Where's the dx in this integral? There is no infinitesimal integrating element in your integral.
     
  6. Aug 26, 2011 #5
    It's called a typo.
     
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