In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the class, 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 notion different. A bounded measurable function can have a non-measurable "area under graph" (at least I believe I can construct it by transfinite induction), so they indeed are not always the same.

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?