In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this way?

The first question is to what extent are the notions different. I believe that a bounded measurable function can have a non-measurable "area under graph" (it should be doable by transfinite induction), but I am not completely sure, so treat it as a part of my question. (EDIT: I was very wrong. The two notions coincide and the argument is very straightforward, see Nik Weaver's answer and one of the comments).

What are the advantages of the Lebesgue integration over area-under-graph integration? I believe that behaviour under limits may be indeed worse. Is it indeed the main reason? Or maybe we could develop integration with this alternative approach?

Note that if a non-negative function has a measurable area under graph, then the area under the graph is the same as the Lebesgue integral by Fubini's theorem, so the two integrals shouldn't behave very differently.

EDIT: I see that my question might be poorly worded. By "area under the graph", I mean the measure of the set of points $(x,y) \in E \times \mathbb{R}$ where $E$ is a space with measure and $y \leq f(x)$. I assume that $f$ is non-negative, but this is also assumed in the standard definition of the Lebesuge integral. We extend this to arbitrary function by looking at the positive and the negative part separately.

The motivation for my question concerns mostly teaching. It seems that the struggle to define measurable functions, understand their behaviour, etc. might be really alleviated if directly after defining measure, we define integral without introducing any additional notions.

come upwith a notion of "area under the graph". This notion does not exist a priori. In reality, we can't geometrically compute areas except of very specific figures, mainly rectilinear ones. As to why we have to define measure theory, perhaps look at the answer on this question: math.stackexchange.com/questions/7436/lebesgue-integral-basics $\endgroup$ – Arturo Magidin Jan 28 '19 at 19:22