What does it mean for an integral's differential to be raised to a power, like e.g. $d^3r$ or $d^3u$? I was reading Zwelfel's book on reactor physics and saw the following notation:



Notice that the differential of the integral is raised to the third power.
What is the meaning of this notation?  Are they trying to imply a volume integral here?  How would I convert this to a Riemann sum?
 A: The notation $\mathrm d^3r$, often also $\mathrm d^3\mathbf r$, is generally understood to indicate a three-dimensional volume integral, as you correctly surmise. If $\mathbf r=(x,y,z)$ then you could also denote that as $\mathrm dx\,\mathrm dy\,\mathrm dz$, or as $\mathrm dV$ if it is clear what the integration variable is. 
The notation $\mathrm d^3\mathbf r$ is more compact and it does a better job at specifying exactly what the integration variable is and what sort of integral is being taken; this is very useful in places where the page is already busy enough, like, say,

where directly specifying the components of $\vec k$ would (i) make the formula much, much longer, and (ii) actually make the text less readable.
In addition to this, it is very common for authors to simply drop the superscript and use notation like $\mathrm d\mathbf r$ when it's clear that it can only be a volume integral (example), and even mix both notations, introducing superscripts when required to indicate the dimensionality of the integral (example).
A: I would normaly understand an Integral $\int d^3\mathbf{r}$
 as a volume integral over the whole space $ \mathbb{R}_3$, where I would understand the bold r there as $\vec{r}$ . I have also seen $\int d^3 \vec{r}$ meaning the same volume integral over $ \mathbb{R}_3$. Or even $\int d~ \vec{r}$ with the superscript dropped (I do not like this one but I have seen it in literature). All the previous expressions are coordinate independent and $\vec{r}$ or bold $\mathbf{r}$ denote the general location vector.
A differential $dr$ without a vector arrow or without the superscript or not in bold is usally (at least in physics) meant as a differential concerning the radius in spherical coordinates. So the volume integral is commonly written as:
$\int d^3 \vec{r} \equiv \int d~ \vec{r} \equiv \int d^3\mathbf{r} = \int_{-\infty }^{+\infty }\int_{-\infty }^{+\infty }\int_{-\infty }^{+\infty }dxdydz=\int_{0 }^{2\pi }\int_{0 }^{\pi }\int_{0 }^{\infty }r^2 \sin(\theta)d\phi d\theta dr $.
Where the last two equalities are in cartesian and spherical coordinates respectively.   
