In synthetic geometry, we simply have an assumption (axiom) that the area of a rectangle with sides $a,b$ is $a·b$. What are lengths in synthetic geometry? It is not specified, except that they form an archimedean field plus closure under geometric construction. If only compass and straightedge constructions are permitted, then the constructible lengths are generated by quadratic extensions on $\mathbb{Q}$. If conics are permitted too, then the constructible lengths are generated by quadratic or cubic extensions on $\mathbb{Q}$. (Even one conic is enough to perform cube-roots.)
In general, synthetic geometry never ever involves lengths beyond algebraic reals. This should not be surprising, because the algebraic reals form a real closed field, which means that it satisfies the same first-order sentences (in the language of ordered fields) as the real numbers. So one can do synthetic geometry using an axiomatization where all lengths form a real closed field (i.e. they are axiomatized to form an ordered field closed under adding roots of nonzero polynomials), and our results will hold whether or not geometric lengths are reals or just algebraic reals.
In modern mathematics, it turns out that we can show the existence of not only a real closed field, but in fact a model of the axiomatization of the real numbers. So we can in fact define Euclidean geometry in terms of Euclidean space $\mathbb{R}^n$, where $n$ is usually $2$ or $3$. The Cantor-Dedekind axiom is a misleading notion; such definition of Euclidean geometry is not an assumption in any sense. The advantage of working with Euclidean geometry is that we can apply tools of real analysis (such as the intermediate-value theorem and differentiation) to solve geometry problems. The disadvantage is that such tools rely on real analysis and so may not apply to synthetic geometry in general.
Anyway, the Riemann/Darboux integral is not defined in terms of any sort of geometric notions! It is simply defined the way it is, and then we can use the integral to define area of regions bounded by curves, and then prove nice properties about that notion of area, such as the area of a rectangle, monotonicity and finite additivity, which any reasonable notion of area ought to have. Indeed, this notion of area is equivalent to Jordan measure. Note that in this approach of defining area we do not define the integral in terms of the area of rectangles! So there is no unproven assumption. By the way, take note that Jordan measure is not countably additive, unlike Lebesgue measure.
