I must find the z centroid of a hemisphere with radius a. It's base is on the x-y plane and its dome extends up the z axis. I am using the following equations to determine the centroid.

$\displaystyle \overline{z}=\frac{\int_V\tilde{z} dV}{\int_V dV}$

I am using $\displaystyle dV=\pi a^2 dz$ and $\displaystyle \tilde{z}=z$ and integrating from 0 to a

$\displaystyle \overline{z}=\frac{\int_{0}^{a} z \pi a^2 dz}{\int_{0}^{a} \pi a^2 dz}$

$\displaystyle \overline{z}=\frac{\pi a^2 \int_{0}^{a} z dz}{\pi a^2\int_{0}^{a} dz}$

$\displaystyle \overline{z}=\frac{ \int_{0}^{a} z dz}{\int_{0}^{a} dz}$

$\displaystyle \overline{z}=\frac{\frac{z^2}{2}|_{0}^{a}}{z|_{0}^ {a}}$

$\displaystyle \overline{z}=\frac{a}{2}$

I know the answer is supposed to be $\displaystyle \overline{z}=\frac{3a}{8}$ Where did I mess up?

Thanks