Suppose I consider three dimensional vectors $x = (x_1,x_2,x_3) \in \mathbb{R}^3$ and I want to represent the volume $V$ comprising the vectors $x$ such that $f(x) \geq g(x)$, for some known functions $f$ and $g$. Can I denote the volume with the following integral? Is the following standard notation?
\begin{equation} V = \displaystyle \int \limits_{\substack{x \in\mathbb{R}^{3}\\ f(x) \geq g(x)}} dx \end{equation}
That should be fine. The only thing that sticks out to me is the use of $dx$ as indicating a integral over $\mathbb{R}^3$. This could be fixed by simply writing $dV$ instead. However, the way I'd do it would be to define our domain of integration as a set first, i.e. $$S = \{x\in \mathbb{R}^3 \mid f(x) \geq g(x)\}$$ And then we can write $$\int_S dxdydz$$