Notation: Why write the differential first?

When you have a lot of integrals, particularly with limits, it can be very helpful at times to be able to tell at a glance which integral is over which variable.

$$\int_0^1 \int_2^3 f(x,y) \; \mathrm d x \mathrm d y$$

This is not particularly readable or clear, especially when $f$ is lengthy and there are more nested integrals etc. I could also imagine it being misinterpreted.

By contrast,

$$\int_0^1\mathrm d y \int_2^3 \mathrm d x \; f(x,y)$$

makes it very clear what is going on. The only price you pay is possible ambiguity about where the integral ends, but this is easier to make clear with formatting and less of an issue anyway.

Edit: It also just occurred to me that the second notation ties in better with the syntax of an operator. That is, if one thinks of $\int_0^1 \mathrm d x$ as being an operator, taking a function to its integral, it's more natural to have the whole operator together in one lump. Think of how one changes $$\frac {\partial f}{\partial x}\to \frac{\partial}{\partial x} f$$

In my opinion, the symbol $\int \mathrm{d}x\, f(x)$ is simply bad notation. On the contrary, when dealing with (at least) double integrals, the advantage of $$\int_{a}^b \mathrm{d}x \int_{c}^{d} \mathrm{d}y\, f(x,y)$$ is that it perfectly fits into the stategy of Fubini's theorem. Imagine that you are computing a double integral: you say "Ok, now I fix the $x$ variable and integrate with respect to the $y$ variable." It is natural to write down $x$ first.

This said, I think that $$\int_{a}^{b} \left( \int_{c}^{d} f(x,y)\, \mathrm{d}y \right) \mathrm{d}x$$ should be used in printed papers and books.

Apart from the idea that $\int_a^b$ should be next to the $\mathrm{d}x$ that makes clear we mean $\int_{x=a}^{x=b}$, or the fact that $\int_a^b\mathrm{d}x$ is a linear operator that deserves a symbol, there are some ways such placement protects us from our own mistakes:

• One advantage of writing $\mathrm{d}x$ first is you won't forget it, which we mustn't do if we have to juggle multiple variables, e.g. due to a substitution. Sometimes, you have to include quite a few terms or factors in the integrand.

• If you're handwriting an integral (which may sound antiquated, but every lecture attendee knows it isn't), writing $\mathrm{d}x$ second carries the further risk of running out of room. You could avoid that by not writing the $\mathrm{d}x$ until you're done, but then you might forget it.