Show that $\int_0^1 dx\int_0^1 f dy=1$, but $\int_0^1 dy\int_0^1 f dx$ does not exists.

Question

If $A=\left\{0\leq x\leq 1;0\leq y\leq 1\right\}$ and $f:A\to \mathbb{R}$ is defined by $$f(x,y)= \begin{cases} 1 & \text{if } x\in \mathbb{Q} \\ 3y^2 & \text{if } x \in\mathbb{Q^c}\end{cases} $$

Show that $\int_0^1 dx\int_0^1 f dy=1$, but $\int_0^1 dy\int_0^1 f dx$ does not exists.

I have solved the first, which is as follows: $\int_0^1 fdy=1$ for both cases, hence $\int_0^1 dx\int_0^1 f dy=1$.

I want to solve the 2nd. Please help me to solve the second.