Center of Mass via integration for ellipsoid I need some help with the following calculation:
I have to calculate the coordinates of the center of mass for the ellipsoid 
$$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 + \left( \frac{z}{c} \right)^2 \le 1,  \quad z \ge 0$$
with mass-density $\mu(x,y,z)=z^2$.
I wanted to use:
$$
\begin{align}
x & = a r \sin\theta \cos\varphi \\
y & = b r \sin\theta \cos\varphi \\
z & = c r \cos\theta
\end{align}
$$
whereas 
$$
\begin{gather}
0 \le r \le 1, \\
0 \le \theta \le \pi, \\
 0 \le \varphi \le 2\pi
\end{gather}
$$
and
$$\frac{\partial (x,y,z)}{ \partial (r, \theta, \varphi)} = r^2 \sin\theta.$$
Did I choose the right things so far?
1) $$
\begin{align}
M & = \int\limits_E µ(x,y,z) d(x,y,z) \\
 & = \int_0^1 \hspace{-5pt} \int_0^{\pi} \hspace{-5pt} \int_0^{2\pi} c^2 r^2 \cos^2\theta \cdot r^2 \sin(\theta) d(r, \theta, \varphi) \\
 & = c^2 \int_0^1 r^4 dr \int_0^\pi \sin\theta \cdot \cos^2\theta d\theta \int_0^{2\pi} d\varphi \\
 & = \frac{4\pi c^2}{15}.
\end{align}
$$
2) $$x_s \cdot M = \ldots $$
Here I get $\int_0^{2pi} \cos\varphi \, d \varphi = 0$, so the whole product is zero, so x_s is zero too??
What am I doing wrong?

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\begin{align}
\color{#00f}{\large\vec{R}_{\rm cm}}&\equiv\left.
{\ds{\int z^{2}\pars{x\hat{x} + y\hat{y} + z\hat{z}}\,\dd x\,\dd y\,\dd z}
\over
\ds{\int z^{2}\,\dd x\,\dd y\,\dd z}}
\right\vert_{{x^{2} \over a^{2}}\ +\ {y^{2} \over b^{2}}\ +\ {z^{2} \over c^{2}}\
<\ 1
\,,\ z\ >\ 0}
\\[1cm]&=
\left.{\ds{\int z^{3}\,\dd x\,\dd y\,\dd z}\over
\ds{\int z^{2}\,\dd x\,\dd y\,\dd z}}\,\hat{z}
\right\vert_{{x^{2} \over a^{2}}\ +\ {y^{2} \over b^{2}}\ +\ {z^{2} \over c^{2}}\
<\ 1
\,,\ z\ >\ 0}
\\[1cm]&=\left.{\ds{\verts{abc^{4}}\int z^{3}\,\dd x\,\dd y\,\dd z}
\over
\ds{\verts{abc^{3}}\int z^{2}\,\dd x\,\dd y\,\dd z}}\,\hat{z}
\right\vert_{x^{2}\ +\ y^{2}\ +\ z^{2}\ <\ 1\,,\ z\ >\ 0}
=\left.{\ds{\int z^{3}\,\dd x\,\dd y\,\dd z}
\over
\ds{\int z^{2}\,\dd x\,\dd y\,\dd z}}\ \verts{c}\,\hat{z}
\right\vert_{r\ <\ 1\,,\ z\ >\ 0}
\\[1cm]&={\ds{2\pi\int_{0}^{1}\dd r\,r^{5}\int_{0}^{\pi/2}\dd\theta\,\sin\pars{\theta}\cos^{3}\pars{\theta}}
\over \ds{2\pi\int_{0}^{1}\dd r\,r^{4}\int_{0}^{\pi/2}\dd\theta\,\sin\pars{\theta}\cos^{2}\pars{\theta}}}\
\verts{c}\,\hat{z}
={5 \over 6}\,{\ds{\int_{0}^{1}\xi^{3}\,\dd\xi}\over
\ds{\int_{0}^{1}\xi^{2}\,\dd\xi}}\ \verts{c}\,\hat{z}
=\color{#00f}{\large{5 \over 8}\,\verts{c}\,\hat{z}}
\end{align}
