Wikipedia has multiple definitions for octant. Some of which are on circles. It would seem to me the easiest version of this question is when octant refers to the cubic region ($x\ge0, y\ge0, z\ge0$), which is also what is explicitly stated on the question. from the 2 equations I can get: $2y^2\le8-2x^2-y^2 \iff y \le \sqrt$ (since $y\ge0$) And that gives me a bounbdary for $y$. but how can I find a boundary for $x$? The only boundary I can come up with would be $0$ to infinity.
$\begingroup$ The first octant refers to the portion of the xyz plane for which $x$, $y$, and $z$ are all positive. (And perhaps the last octant is the portion of the xyz plane for which $x$, $y$, and $z$ are all negative.) As far as I know, there's no standard convention for naming the octants other than the first and last. In other words, if I say "look at the second octant," one can't actually be sure of what I'm talking about. $\endgroup$