My textbook says the following when discussing continuous random variables and probability density functions:
A probability density function $p(x)$ does not give the probability of a specific state directly; instead the probability of landing inside an infinitesimal region with volume $\delta x$ is given by $p(x) \delta x$.
My understanding is that, in the univariate case, it should be "area" not "volume, right? Wouldn't it only be "volume" in the multivariate case?
While it's technically correct to call it a volume (this term is not restricted to three- or even multi-dimensional space in mathematics), judging from the context it's indeed a bad choice of terminology here, since the entire surrounding text treats $x$ as a univariate variable and uses two separate variables $x$ and $y$ in all two-dimensional examples. If $x$ were known to be univariate, “length” or “measure” would be a much more usual choice of term; “volume” would still be technically correct but unnecessarily confusing. Using “volume” here seems like a half-hearted attempt to generalize; while the generalization is in order, it would have required either more explanation or a notation that distinguishes the implied multivariate from the surrounding univariates.