Doing homework, I came across a question like the following: $$ \int_{[-1,1]\times[0,1]}\sin(x^2y^3)\,\mathrm dx\,\mathrm dy. $$ I am having trouble understanding what the notation means? Does it simply represent $$ \int_{-1}^{1}\int_{0}^{1}\sin(x^2y^3)\,\mathrm dx\,\mathrm dy? $$
I am assuming this is referring to set notation?
On
Kind of. The integral over $[-1, 1] \times [0, 1]$ is a double integral, which has an analogues definition to the ordinary integral, though it uses partitions consisting of 2D regions rather than subintervals, but by Fubini's Theorem, it is equal to the iterated integral you suggest (if you calculate the integrals in the right order).
In this context, the notation $A\times B$ represents the Cartesian product, which is defined as $$ A\times B=\{(a,b)\mid a\in A\ {\mbox{ and }}\ b\in B\}. $$ So for your problem, $$ [-1,1]\times [0,1]=\{(x,y)\mid x\in [-1,1]\ {\mbox{ and }}\ y\in [0,1]\}, $$ which leads us to write $$ \iint_{[-1,1]\times[0,1]}(\cdots)\,\mathrm dx\,\mathrm dy =\int_0^1\int_{-1}^1(\cdots)\,\mathrm dx\,\mathrm dy. $$