What is wrong in turning this expression in spherical coordinates :
$$\int_{0}^{\pi}\int_{0}^{2\pi}\vec{r}\, \cos \Theta \, \sin \Theta \, \mathrm{d} \Theta \, \mathrm{d} \phi $$
to this :
$$\int_{0}^{\pi} \cos \Theta \, \sin \Theta \, \mathrm{d} \Theta\int_{0}^{2\pi}\vec{r}\, \, \mathrm{d} \phi = 0 \cdot \int_{0}^{2\pi}\vec{r}\, \, \mathrm{d} \phi = 0 $$
In other words, why can't I pull out all the $\theta$ terms that are constant relative to the $\phi$ integration ?