The formula for the area is: $$\iint_RdA$$
The formula for the volume is: $$\iint _R (f(x,y)-g(x,y))\,dA$$
In other words, the volume is surface base's area * height, or the base repeated over and over throughout a certain height, if I understand this correctly. Which I think implies that solids have to be decomposed in cubic/parallelepiped/cylindrical like shapes.
What's confusing me is that in the formula for the area is that it can be rewritten as: $$\iint_RdA = \iint_R (1-0)\,dA$$
in other words, a solid with 1 of height. Which means it's giving a volume, not an area. What am I getting wrong here? I also don't see where the Riemann-like approximation with the parallelepipeds fits in this.