Error in dipole moment calculation

[Gavroy]

I am calculating the dipole moment of a metal sphere in a uniform electric field $E_0$ in z-direction. From here I know that the charge density [look at page 15][1]


http://www.phy.iitb.ac.in/~dkg/PH-102/conductors.pdf[/PLAIN]

is given by $3 \epsilon_0 E_0 cos(\theta)$

Now I wanted to calculate the resulting dipole moment by $\int_0^R \int_0^{2\pi} \int_0^{\pi} r^3 sin(\theta) cos(\theta) 3 \epsilon_0 E_0 d\phi d\theta dr$

but in this case the integral over $\theta$ is zero and therefore this whole term will be zero which is somewhat strange, since there should be a dipole moment. So what am I doing wrong?

 

[mfb]

One factor of r in your r³ is a vector, not the magnitude of r. It depends on θ.