- #1
P3X-018
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Homework Statement
The problem is this:
Let f : R^2 -> R be
[tex] f(x,y) = \frac{xy}{(x^2+y^2)^2}\qquad (x,y)\neq(0,0) [/tex]
With [itex] f(0,0) = 0 [/itex].
Question:
Is [itex] f [/itex] integrable with respect to the Lebesguemeasure [itex] m_2 [/itex] om the set [itex] [-1,1]\times[-1,1] [/itex]?
The Attempt at a Solution
Well [itex]f[/itex] is integrable if and only is
[tex] \int_{[-1,1]^2} \vert f\vert\,dm_2(x,y) < \infty[/tex]
The integral is infinite, and so f isn't integrable.
The thing that makes it confusing, is that in the next problem it says,
Show that both the double integrals exist
[tex] \int_{[-1,1]}\left(\int_{[-1,1]} f(x,y)\,dm(x)\right)dm(y)[/tex]
[tex] \int_{[-1,1]}\left(\int_{[-1,1]} f(x,y)\,dm(y)\right)dm(x)[/tex]
But how can they exist if f isn't integrable?
EDIT:
Hmm I had made an error. It did seem kinda wired that the integral of a positive would give me something like [itex] \infty-\infty[/itex]. The integral is infinite though. The expression I had found would only hold for y > 0.
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