Actually, in the following book the Lebesgue integral is defined the way you suggested:

**Pugh, C. C.** *Real mathematical analysis*. Second edition. Undergraduate Texts in Mathematics. Springer, Cham, 2015. 

First we define the planar Lebesgue measure $m_2$. Then we define the Lebesgue integral as follows:

> **Definition.** The *undergraph* of $f:\mathbb{R}\to[0,\infty)$ is $$ \mathcal{U}f=\{(x,y)\in\mathbb{R}\times [0,\infty):0\leq y<f(x)\}. $$ The
> function $f$ is *Lebesgue measurable* if $\mathcal{U}f$ is Lebesgue measurable
> with respect to the planar Lebesgue measure and then we define $$
\int_{\mathbb{R}} f=m_2(\mathcal{U}f). $$

I find this approach quite nice if you want to have a quick intoroduction to the Lebesgue integration. For example: 

> You get the monotone convergence theorem for free: it is a
> straightforward consequence of the fact that the measure of the union
> of an increasing sequence of sets is the limit of measures.