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> $f:[0,1]\times [0,1]\to\mathbb R,$ defined by $$f(x,y)= \begin{cases}1,\quad \ \ y\in\mathbb R\text{\\}\mathbb Q\\2x,\quad\text{otherwise}\end{cases}$$.
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$1.1$: $\int\_0^1f(x,y)dx$ exists for every $y\in[0,1]$ and is equal to $1$.

$1.2$: The iterated integral $\int\_0^1(\int\_0^1f(x,y)dx)dy$ exists and is $1$. 

$1.3$: The double integral $\int\_If(x,y)d(x,y)$ does not exist. 

I am struggling with solving iterated integrals in general and with this one I don't even know where to start since the values kind of jump from 1 to 2x constantly. 

Edit: Got an idea for 1.1.: I made two cases, one for an irrational y and one for the rest. 
Giving me $\int\_0^11dx$ which is 1 and $\int\_0^12xdx$ which also is 1. 

Could someone give me a short explanation about them and some hints on how to approach these exercises?