Uguaglianza tra integrali doppi

[fabioz96]

Salve,

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

Grazie

[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