• $\displaystyle{\iint_R f(x,y) \, dA}$ is the double integral of $f(x,y)$ over the region $R$.


• $\displaystyle{\iint_R f(x,y) \, dx\, dy}=\displaystyle{\iint_R f(x,y) \, dy \, dx}=\displaystyle{\iint_R f(x,y) \, dA}$.
  

You can see this by noticing the area of the small rectangle is $\Delta A = \Delta x \,\Delta y = \Delta y\, \Delta x$, so the infinitesimal area is $dA = dx\, dy = dy\, dx$.

• $\displaystyle{\int_a^b \int_c^d f(x,y)\, dy \, dx}$ is an iterated integral. We first treat $x$ as a constant and integrate $f(x,y) \,dy$ from $y=c$ to $y=d$.  Call the result $g(x)$.  It is a function of $x$ and describes the volume of the column with that value of $x$.  We then integrate $g(x)\, dx$  to get the volume of all columns from $x=a$ to $x=b$.


• $\displaystyle{\int_c^d \int_a^b f(x,y)\, dx \, dy}$ is an iterated integral.  We first treat $y$ as a constant and integrate $f(x,y) \,dx$ from $x=a$ to $x=b$.  Call the result $g(y)$.  It is a function of $y$ and describes the volume of the row with that value of $y$.  We then integrate $g(y)\, dy$  to get the volume of all rows from $y=c$ to $y=d$


• In an iterated integral, the order of $dx$ and $dy$ tells you which variable to integrate first. When in doubt, draw parentheses: $$\int_a^b \int_c^d f(x,y) \, dy\, dx = \int_a^b \left ( \int_c^d f(x,y) \,dy \right )\, dx.$$


• Fubini's Theorem says that you can evaluate double integrals by doing an iterated integral in either order, but sometimes one order is a lot simpler than the other.