Notation and order

• $\displaystyle{\iint_R f(x,y) \, dA}$ is the double integral of $f(x,y)$ over the region $R$.
• $\displaystyle{\iint_R f(x,y) \, dx\, dy}$ and $\displaystyle{\iint_R f(x,y) \, dy \, dx}$ mean the exact same thing as $\displaystyle{\iint_R f(x,y) \, dA}$. The area of a little box 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 how much stuff is in the column with that value of $x$. We then integrate $g(x) dx$ from $x=a$ to $x=b$.
• $\displaystyle{\int_c^d \int_a^b f(x,y) dx \, dy}$ is an iterated integral where we first integrate over $x$ to get the contribution of a row, and then integrate over $y$ to add up all the rows.
• 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.

In the following video, we go over notation (2 minutes) and work two example problems (4 minutes each). The first example involves integrating $6x^2y$ over the rectangle $[1,3] \times [0,2]$, and both orders of integration are equally easy. The second example involves integrating $x e^{xy}$ over the same rectangle, and one order turns out to be much easier than the other.