Why isn't integral defined as the area under the graph of function?

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 introduction 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.

As pointed out by Nik Weaver, the equality $\int(f+g)=\int f+\int g$ is not obvious, but it can be proved quickly with the following trick: $$ T_f:(x,y)\mapsto (x,f(x)+y) $$ maps the set $\mathcal{U}g$ to a set disjoint from $\mathcal{U}f$, $$ \mathcal{U}(f+g)=\mathcal{U}f \sqcup T_f(\mathcal{Ug}) $$ and then

$$ \int_{\mathbb{R}} f+g= \int_{\mathbb{R}} f +\int_{\mathbb{R}} g $$

follows immediately once you prove that the sets $\mathcal{U}(g)$ and $T_f(\mathcal{U}g)$ have the same measure. Pugh proves it on one page.

If $f: \mathbb{R} \to [0,\infty)$ is Borel (or Lebesgue) measurable, then for each rational $a > 0$ define $X_a = f^{-1}([a,\infty)) \times [0,a)$. Then each $X_a$ is measurable and their union is exactly the region under the graph. So the region under the graph is measurable.

I think the reason why we develop the Lebesgue integral in the usual way is because it provides a powerful technique (characteristic functions --> simple functions --> arbitrary measurable functions) for deriving the basic theory of the integral. Even simple things like $\int f + \int g = \int (f + g)$ aren't obvious if you take "measure of the region under the graph" as the definition.

The "area under a graph" approach is used in Wheeden/Zygmund's 1977 text Measure and Integral. An Introduction to Real Analysis, a book that was used (among other possible places) in the early 1980s for a 2-semester graduate real analysis course at Indiana University (Bloomington).

(second sentence of Chapter 5, on p. 64) The approach we have chosen [for the integral of a nonnegative function $f:E \rightarrow [0, +\infty],$ where $E \subseteq {\mathbb R}^n$ is measurable] is based on the notion that the integral of a nonnegative $f$ should represent the volume of the region under the graph of $f.$

I looked in the preface and elsewhere for any historical or literature citations about this approach and did not see anything relevant. Also, later in this book abstract measure and integration theory is developed in one of the standard ways.