I am given the following multivariate calculus question:
Integrate $f(x,y,z)=14xz$ over the region in the first octant $(x,y,z \ge 0)$ above the parabolic cylinder $z=y^2$ and below the paraboloid $z=8-2x^2-y^2$.
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:
$2y^2\le8-2x^2-y^2 \iff y \le \sqrt{4-x^2}$ (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.
$x$ ranges from $0$ to $2$: The octant constraint gives the $0$. As for the $2$: it should be apparent from what you've done that the region (projected to the $xy$ plane) should be a quarter circle:
Because of course, when $x>2$, the constraint on $y$ becomes undefined.
The final integral's bounds are:
$$\int_0^2\int_0^{\sqrt{4-x^2}}\int_{y^2}^{8-2x^2-y^2}(\cdots)dzdydx.$$
Does this make things clearer?