I need to evaluate the triple integral of the function $$f(x,y,z) = z$$ over the area/body $Q$ (where $Q \subset \mathbb{R}^3$) which is enclosed by the ellipsoid
$x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$
Here $a,b,c$ are positive constants.
I am getting zero and to me it kind of makes sense because this body is so to speak symmetrical w.r.t. $z$. I mean when the point $(x_0, y_0, z_0)$ is in $Q$, so is $(x_0, y_0, -z_0)$.
But my book is giving this answer: $abc^2 \cdot \pi/4$
Which one is correct?
Did they mean in the book to just integrate over this part of Q which is above the Oxy plane? Or am I missing something else?