# Denoting a volume integral

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?

$$$$V = \displaystyle \int \limits_{\substack{x \in\mathbb{R}^{3}\\ f(x) \geq g(x)}} dx$$$$

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 $$\int_S dxdydz$$