Double integral notation

Question

Over a region D (a bounded, closed and connected region), can we write the double integral $\iint\limits_D \, f(x,y)\,dx\,dy$ as $\iint\limits_D \, f(x,y)\,dy\,dx$ (note the order of $dx$ and $dy$)?

Answer 1

If either:

• $f\ge 0$;
• $\iint\limits_D |f(x,y)| dxdy<\infty$;
• $\iint\limits_D |f(x,y)| dydx<\infty$,

you can. This is true if, for example, $f$ is bounded.

Answer 2

In my opinion $$ \iint_D f(x,y)\,\mathrm dx\,\mathrm dy$$ for $D\subseteq\mathbb R^2$ is bad notation. It suggests that what you are looking at is an iterated integral, where you integrate with respect to $x$ first and afterwards with respect to $y$. But it is not, it's an integral over a region in $\mathbb R^2$. You are integrating with respect to the two-dimensional Lebesgue-measure. Better notations are $$ \iint_D f(x,y)\,\mathrm dA\ , \int_D f(x,y)\,\mathrm dA\ , \int_D f\,\mathrm d\lambda $$ or variants of that. Those notations don't suggest an iterated integral.

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If we really wanted to interpret $\iint_D f(x,y)\,\mathrm dx\,\mathrm dy$ as an iterated integral, it would be $$ \int_{\operatorname{pr}_2(D)} \left( \int_{\{x \in\mathbb R \mid (x,y)\in D\}} f(x,y)\,\mathrm dx \right) \mathrm dy $$ where $\operatorname{pr}_2$ is the projection $(x,y)\mapsto y$ and $\{\,x\in\mathbb R \mid (x,y)\in D\}$ are all the possible $x$-values in the horizontal slice of $D$ where the second coordinate equals a fixed $y$. Hiding this behind $\iint_D$ is another bad idea.