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.