I have been informed that consecutive differentials in iterated integral problems are actually connected via the exterior product. So the factor $dx\ dy$ in $\int\int x^2\ dx\ dy$ is actually the quantity $dx\ \Lambda\ dy$. How is this consistent with the notion of this integral as the summation of tiny squares of area, $dA$, which should instead be the product of $dx$ and $dy$ under scalar multiplication?
Also, why doesn't switching the order of integration from $dx\ dy$ to $dy\ dx$ or vice-versa negate the answer to the double integral, since $-dx\ \Lambda\ dy = dy\ \Lambda\ dx$?
Without making a big elaborate deal out of this, once you choose the standard orientation on $\Bbb R^2$, the definition of $\int_R f\,dx\wedge dy$ is the double integral $\int_R f\,dA = \iint_R f\,dx\,dy = \iint_R f\,dy\,dx$ (where you put appropriate limits on the iterated integrals). Yes, $dy\wedge dx = -dx\wedge dy$, but the rule is that you must put things in the correct order according to the orientation of your manifold (region). This is, admittedly, always a confusing point for those of us first learning to integrate differential forms.