If we wish to do a nested integration over a function of multiple variables, is there a (or, what is the) correct order to write the limits on the integral symbol?
For example, to integrate the function $f(x,y)$ first with respect to $x$ over the range $(x=-1$ to $x=1)$, and then secondly with respect to $y$ over the range $(y=-2$ to $y=2)$, which of the following is the correct notation:
Option 1:
$$
\int_{-2}^{2} \int_{-1}^{1} f(x,y)\; dx\;dy
$$
Option 2:
$$
\int_{-1}^{1} \int_{-2}^{2} f(x,y)\; dx\;dy
$$
Are the integration symbols "nested", starting with the inner ones and working outwards (as in option 1)? Or, do they maintain the order as the differential operators (as in option 2, i.e the $x$ one comes first, then the $y$ one).
Best Answer
Option 1 is the only convention I know (except the one kimchi wrote in the comments, but this is non of your two options and only used by physicists, as far as I know). When you write $$ \int_{-2}^{2} \int_{-1}^{1} f(x,y)\; dx\;dy $$ you can imagine brackets like in $$ \int_{-2}^{2}\left( \int_{-1}^{1} f(x,y)\; dx\right)\;dy $$ which are usually not written.