I'm talking about the integral part that is highlighted:
Should I interpret the top one highlighted as the upper bound of integration and the bottom one as the lower bound? That's the only interpretation that made sense to me but I'm not sure. Thanks.
Since you can't order vectors in a geometrically sensible way, multiple integrals don't really have upper and lower bounds; they just have a set over which the integration occurs. Single integrals can be thought of in these terms too; if $a<b$, $\int_a^bf(x)dx=\int_{x\in[a,\,b]}f(x)dx$. (And you can integrate over other sets too, e.g. $\int_{[0,\,1]\cup[2,\,3]}f(x)dx=\int_0^1f(x)dx+\int_2^3f(x)dx$.) In your integral, the highlighted conditions define the set of values for $x\in\Bbb R^n$ over which we integrate.