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Calculate the Lebesgue integral of the function

$$ f(x,y)=\left\lbrace\begin{array}{ccl}[x+y]^{2} &\quad&|x|,|y| <12 ,\quad xy \leq 0\\ 0 &\quad&\text{otherwise}\end{array} \right.$$

in $\mathbb{R}^2$.

Can anyone help with this? I can't find a way to make the expression of $f$ more simply to calculate the integral.

edit: $[\cdot]$ is the integer part.

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    In general, we use $\lfloor \cdot \rfloor$ to denote the floor function. That's "\lfloor" and "\rfloor".2012-01-22
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    This function has finite range. It can be integrated by drawing a picture.2012-01-22
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    @ncmathsadist: Can you explain this a little more?2012-01-22
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    The integrand breaks up into a finite number of cases. It is piecewise constant on strips between lines of the form $y = a - x$ and $y = a + 1 - x$, where $a$ is an integer. Draw the slices; the function is constant between the parallel lines that result. Multiply the area of each strip by the value of the function on the strip.2012-01-22
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    The Riemann integral gives the same value (why?)2012-01-22
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    @AD.: What's the value of the integral?2012-01-22

2 Answers 2

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Denote $$ A_{m,n}=\{(x,y):m\leq x

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Hint:

  1. The function is non-negative, and hence one may apply Tonelli's theorem (sometimes cited as Fubini-Tonelli's or even Fubini' theorem).

  2. Draw the domain of integration (that is the set where $f(x,y)\ne0$). Split up the domain in order to adopt step 1.

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    Кого я вижу! Ты ушел с dxdy?2012-01-22
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    @Norbert What do you mean?2012-01-23
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    May be I've confused you with my friend from MSU. Artem is it you?2012-01-23
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    @Norbert Sorry that is not me. :)2012-01-23
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    What a nuisance... The reason of this confusion is that he have the same nick and the same range of mathematical interests as yours!2012-01-23
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    @Norbert Sounds like a nice fellow!2012-01-23