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Homework Help: Henstock Integral Help

  1. Apr 22, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove [itex]\int\int_{[-1,1]×[-1,1]}f(x,y)dA[/itex] is not Henstock Integrable.

    2. Relevant equations
    [itex]f(x,y) = \frac{xy}{(x^{2}+y^{2})^{2}}[/itex]
    [itex]f(x,y) = 0 [/itex] if [itex]x^{2}+y^{2}=0[/itex] on the region [-1,1]×[-1,1]

    3. The attempt at a solution
    The only hints given is that we will not be able to solve in Maple or Mathematica by using trapezoidable interagtion techniques.

    I'm assuming I would start by trying to prove that it is integrable and use contradiction to solve my answer.
    Last edited: Apr 22, 2013
  2. jcsd
  3. Apr 22, 2013 #2


    Staff: Mentor

    What does it mean to be Henstock integrable? That's not a term I've ever heard.

    [STRIKE]In any case, your second equation above doesn't apply. x2 + y2 ≠ 0 for any real x and y.[/STRIKE]
    Edit: Wrote the above without thinking too clearly.
    Last edited: Apr 22, 2013
  4. Apr 22, 2013 #3
    Henstock-Kurzweil Integral is used in Real Analysis. I'm pretty sure my class is being taught material that is generally left for graduate school, however since my professor did his PhD studies on it, thinks we can handle it.

    As for the second equation, we are to think of f(x,y) as a piecewise function. That way we can deal with when both x and y are zero at the same time.
  5. Apr 22, 2013 #4


    Staff: Mentor

    Sorry about that incorrect comment in my previous post. I was thinking x2 + y2 ≥ 0, and that somehow that meant that x2 + y2 ≠ 0.
  6. Apr 22, 2013 #5
    I think it's clear the problem is in 0. Try to find out what the integral is on ##([-1,1]\times[-1,1])\setminus ([-\varepsilon,\varepsilon]\times [-\varepsilon,\varepsilon])## and let ##\varepsilon\rightarrow 0##.
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