Is that integral $\iiint\limits_{G}\frac{dxdydz}{(x^2+2y^{2}+3z^{2})^{3/4}}$ finite? Let $$G=\{(x,y,z)\in \mathbb{R}^{3}:x^{4}+y^{6}+z^{8}<10\}$$
We want to check if $\iiint\limits_{G}\frac{dxdydz}{(x^2+2y^{2}+3z^{2})^{3/4}}$ is finite. My first idea is to try to calculate integral on surface $B(0,\epsilon)$, for random $\epsilon$ and find the $\lim_{\epsilon \to 0^{+}}\iiint\limits_{B(0,\epsilon)}\frac{dxdydz}{(x^2+2y^{2}+3z^{2})^{3/4}}$, but i'm not sure about this. Is my claim right and if not how can we deside if integral is finite or not ?

You can save yourself a ton of trouble by transforming the integral. If the denominator were $x^2+y^2+z^2$, then I would suggest to switch to polar coordinates, see Example 3 here:
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
This is a little bit different because you have to apply a linear transformation first (using the same principle as in Example 3, so compute the Jacobian).
You need not worry about the borders. Just determine if the integral is finite over the domain $[-\sqrt[4]{10}, \sqrt[4]{10}]^3$, the important stuff happens around the origin anyway.
