What is the meaning of $dA$ in $\iint_E\dots dA$, where $E$ is a region in the $xy$ plane?
In some integrals we use $dA=dx\,dy$, but in others $dA=\hat{k}\,dx\,dy$. (Here $\hat {k}$ is the unit vector in the $z$ direction.)
Why this difference? Also, is there a general formula for $dA$?
Consider a vector field $\mathbf v$. Say we want to compute the flux through some oriented surface $\mathcal S$. What will our flux integral look like? $$\iint_\mathcal{S} \ldots $$ Well, our vector field will be like a weight function by which we integrate our surface. We accomplish this with the inner product $\mathbf v \cdot \hat{\mathbf N} \mathrm{dA}$, where $\hat{\mathbf N}$ is normal to the surface and $\mathrm{dA}$ is the surface area element.
In this case one could choose the notation $\mathrm d \mathbf A = \hat{\mathbf N} \,\mathrm{dA}$, to mean the vector surface area element.
So, we need to distinguish a surface area element $\mathrm{dA}$, which could be $\mathrm{dx\,dy}$, from a vector surface area element $\mathrm{d \mathbf A}$, which has an orientation.