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Is the following statement:

$\int_{x_1}^{x_2}{\int_{y_1}^{y_2}{g(x)+h(y)dxdy}} = \int_{x_1}^{x_2}{g(x)dx} + \int_{y_1}^{y_2}{h(y)dy} $

False in the general case ? Does it hold if $g(x) =|x|$ and $h(y) = |y| $ ?

Thanks in advance

EDIT : answered

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up vote 0 down vote accepted

In general that's false, but a similar result is true:

$ \begin{align} \int_{x_1}^{x_2}\int_{y_1}^{y_2}g(x)+h(y)\ dxdy &= \int_{x_1}^{x_2}\int_{y_1}^{y_2}g(x)\ dxdy + \int_{x_1}^{x_2}\int_{y_1}^{y_2} h(y)\ dxdy \\ &= (y_2-y_1)\int_{x_1}^{x_2}g(x)\ dx + (x_2-x_1)\int_{y_1}^{y_2}h(y)\ dy \end{align} $

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thank you, now I see it. – baibo May 3 '13 at 19:21
    
@JavierBadia Consider giving a proof for both equalities. – ThisIsNotAnId May 3 '13 at 19:42
    
@ThisIsNotAnId: Both follow from the linearity properties of the integral. I didn't include a proof because I thought they were pretty straightforward. – Javier May 3 '13 at 20:35

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