Actually, in the book:
Pugh, C. C. Real mathematical analysis. Second edition. Undergraduate Texts in Mathematics. Springer, Cham, 2015.
The Lebesgue integral is defined the way you suggested. 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 $$ Uf=\{(x,y)\in\mathbb{R}\times [0,\infty):0\leq y<f(x)\}. $$ The function $f$ is Lebesgue measurable if $Uf$ is Lebesgue measurable with respect to the planar Lebesgue measure and then we define $$ \int_{\mathbb{R}} f=m_2(Uf). $$