I want to calculate the volume of the solid defined as follows.
$$ K := \left\{ (x,y,z) \in \Bbb R^3 : x^2 + y^2 + 3z^2 \le 4, z > 0 \right\} $$
I know that this is a half solid ellipsoid and that the volume of an ellipsoid is calculated by $\frac{4}{3}\pi abc$. Thus, for the whole ellipsoid, we should get the volume $\frac{32\pi}{3\sqrt{3}}$ but should I then divide this by $2$ as well since it is given that $z>0$ on $K$?
I saw a similar task where this was not done and I don't know if it is me who is wrong about dividing it by two, or if a mistake was made in the solution to the task.