All the other answers that say the "single" integral is simply a shorthand notation are right, but it is well to remember that one can indeed construe the integral as a single integral as a Lebesgue integral (if you do nothing else, look up Lebesgue's very cute little half paragraph summary (on the Wiki page) of his idea in a letter to Paul Montel).
If you've not met this before, the idea is to define a function that assigns a measure (of "how big") to certain well behaved subsets of the two (or any dimensional region). Then one constructs the sum of the measures, weighted by $f$, of all the subsets which are preimages of intervals $(f,\,f+\Delta f)$ (i.e. subsets wherein the value of $f$ lies between certain values) and passes to certain bounds on these sums as $\Delta f$ is allowed to take any positive value to define the integral.
One of the upshots of this approach is that the domain of integration (in this case a two dimensional plane) does not have to be thought of as slices of lower dimensional spaces pasted together, thus leading to iterated, multiple integrals which one must tackle with the Riemann integral. One just thinks of a lone, non-iterated sum of measures of subsets weighted by the integrated object (in this case $f$). So in the Lebesgue sense $d\,A$ is simply the measure function defined for two-dimensional subsets of the plane.
However, Lebesgue integration over multidimensional manifolds can also be construed as an iterated, in this case, double integral, just like the Riemann integral is and Fubini's theorem is, given certain conditions on $f$, essentially that the iterated Lebesgue integral equals the one defined thinking of subsets of the plane as atomic entities: a corollary of which is that the order of the integration is not important (because the iterated integrals in any order are all equal to the single, multidimensional one).
Of course the two concepts of integral, Lebsegue and Riemann, co-incide given certain conditions on the integrated function. The Lebesgue is more general: Lebesgue integrals are always defined and equal to the Riemann integral when the latter exists, but the converse is not true (there are functions that have a Lebesgue integral, but no Riemann one). Lebesgue integration is extremely important for probability theory, where it is not always convenient to break down an "event" (i.e. measurable subset of the event space) into unions of "lower dimensional" events.