Explanation of how this Lebesgue integral is 1

Question

Let $f(x) =$

$x$ if $x \in \mathbb{Q}$,

$1$ is $x \notin \mathbb{Q}$

The $f(x) = x$ if $x \in \mathbb{Q}$, will give infinity many 'heights' for rectangles against the function. And these heights that will approach infinity themselves. Although the width for each rectangle will be infinitesimally small...so how does this resolve to a value of $1$?