I want to find a center of the mass of the upper part of a homogenous ellipsoid: ${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2} \le 1$, where $a,b,c > 0$. Because of symmetry, I need to find only the 'z' component. Is it all right what I try to do?
$\rho$ is a density
$\bar{z}={1 \over m} \displaystyle\int_D z \rho(\vec{v})d\vec{v}={1 \over V}\displaystyle\int_{-a}^a \bigg[\displaystyle\int_{-b \sqrt{1- {x^2 \over a^2}}}^{b \sqrt{1- {x^2 \over a^2}}}\! \bigg[\displaystyle\int_{0}^{c \sqrt{1-{x^2 \over a^2}-{y^2 \over b^2}}}\! z \ dz \bigg]dy\bigg]dx $