In the original, I need to calculate this multiple integral: $\iiint_V z \ dx \ dy \ dz$, where $V$ is defined by a surfaces: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \le 1, \quad z \ge 0. $$
It's easy to see, that the first inequality is a ellipsoid and the second one means, that we're taking only a half of it, where $z$ coordinate is non-negative.
I think, in this case, it's a good idea to use spherical coordinates as a substitute: $$ \begin{cases} x = a \cdot r \cdot \cos(\phi) \cos(\psi) \\ y = b \cdot r \cdot \sin(\phi) \cos(\psi) \\ z = c \cdot r \cdot \sin(\psi) \end{cases} $$
The Jacobian is equal to $a \cdot b \cdot c \cdot r^2 \cdot \cos(\psi)$, so the integral transforms to $$ \iiint_{\Omega} (c \cdot r \cdot \sin(\psi)) \cdot (a \cdot b \cdot c \cdot r^2 \cdot \cos(\psi)) \ dr \ d \phi \ d \psi $$ and here I'm stuck. How can I calculate integration limits? The only thing I can notice, that $r \leq 1$, just cause ellipsoid in this coordinates looks like $r = 1$.
Hint: except for the factors $a,b,c$, it's the same integration that you would do for the volume of a hemisphere.