homework and exercises - Understanding the integral for the electric dipole moment of a charge distribution

In problem 3.35 of Griffiths' Introduction to electrodynamics, he states:

A solid sphere, radius $R$, is centered at the origin. The “northern” hemisphere carries a uniform charge density $\rho_0$, and the “southern” hemisphere a uniform charge density $−\rho_0$. Find the approximate field $E(r,θ)$ for points far from the sphere ($r \gg R$).

The dipole moment is by definition

$$\textbf{p}=\iiint \textbf{r}'\rho(\textbf{r}') \;\mathrm{dV}$$

But Griffiths uses $z=r'\cos \theta$ and says

$$\textbf{p}=\iiint \textbf{z}\rho(\textbf{r}') \;\mathrm{dV}$$

How does this work? Aren't you supposed to use $r'$ in the integral?

In my calculations I get

$$\textbf{p}= \iiint_\text{northern hemisphere} \textbf{r}'\rho_0 \;\mathrm{dV} -\iiint_\text{southern hemisphere} \textbf{r}'\rho_0\;\mathrm{dV}$$

which gives $$\textbf{p}=0$$

when evaluated, which is wrong. Where have I setup my integral wrong?