Evaluate integral inside integral

First, sketch the domain of integration based on the first formula:

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Then let's see what happens if we want $y$ to be the integration variable in the outer integral. Obviously we will have to integrate from $-\infty$ to $+\infty$ with respect to $y$.

The question remains: given an $y$ (blue lines) what is the domain of integration with respect to $z$?

It is easy to see that if $y<x$ then the domain is $(-\infty,y)$ if, however, if $y>x$ then the domain is $(-\infty,x)$. So, the domain is $(-\infty,\min(x,y)).$

Tags:

Calculus

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