Uguaglianza tra integrali doppi

fabioz96
Salve,

sapresti dirmi la dimostrazione di questa uguaglianza? Oppure indicarmi dove posso trovarla?

Grazie

Risposte
javicemarpe
From the first integral we know that the domain in which you are integrating is (green) $\Omega=\{(x,u)\in\mathbb{R^2}:u\le x, x>0,u>0\}$. Then, if you want to change the order of the integrals, you have to change the way you cover $\Omega$. As you can see in the picture, what you have to do is to fix $u>0$ (red) and, then, cover the corresponding horizontal (blue) line in $\Omega$, which is the set $x\ge u$ (and that's why the second integral is $\int_{u=0}^\infty\int_{x=u}^\infty$, because you fix $u>0$ (red) in the first integral, and then you cover the horizontal (blue) line $x\ge u$ in the second one).

fabioz96
Thank you so much. Now I really understand :)

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