So i have region G: Let $$\alpha , \beta , \gamma \in \mathbb{R^+}$$ $$(\alpha x^2+ \beta y^2+ \gamma z^2) \leq z\sqrt{\alpha x^2+\beta y^2}$$
And i want to solve this integral: $$\iiint\limits_G \sqrt{\alpha x^2+\beta y^2}{dV}$$
So for $\alpha=\beta=\gamma=1$ i drawn the equation, so without the "leq" sign with only "=" in wolphram, seems like something similar to a sphere, so i think i will have to use spherical coordinates, but a little changed for constants $\alpha,\beta,\gamma$. So they will be much easier to solve in the integral part of the task.
I wonder how can i do that, any help with this would be appreciated, i know i didn't show much work, but i have problems making the right coordinates.
Do a change of variables: $$X=\sqrt\alpha x\\Y=\sqrt\beta y\\Z=\sqrt\gamma z$$ With these, $dV=dXdYdZ/\sqrt{\alpha\beta\gamma}$ and your region $G$ is given by $$X^2+Y^2+Z^2\leq Z\sqrt{X^2+Y^2}/\sqrt{\gamma}$$ Now you should be able to use spherical coordinates