I am a little confused what difference is there when it comes to first and double integrals when dealing with surfaces. For example if have to find $$\int_S (2xy\textbf{i} + yz^2\textbf{j} +xz\textbf{k}) \,dS$$
were the region is bounded by x = 0, y = 0, y = 3, z=0 and x + 2z = 6.
So if the divergence theorem is utilized, by simply finding the div(F) and the volume of the whole region, this integral could be worked out.
So why exactly, the divergence theorem is always stated as:
$$\iint_S \textbf{F}.\textbf{N}dS = \iiint_V div(\textbf{F})dV $$
Where here was have a double integral instead of a single integral of the whole surface?