# How to define boundaries for triple integration

I've been struggling when I need to find the boundaries of integration for non-trivial cases. When I do an exercise where the functions are bound by a box or a sphere, I can get the limits easily enough. But on other cases, I'm having a hard time understanding or finding a method to determine those values.

My current case: I need to evaluate $$\iiint_{Q}{f(x, y, z)dV}$$ where Q is the region in the first octant limited/bounded by the coordinated planes and by the graphs $$z-2=x^2+\frac{y^2}{4}$$ and $$x^2+y^2=1$$

Now, since the problem mentions the first octant, I'm guessing $$0\leq x$$, $$0\leq y$$ and $$0\leq z$$. But how do I move from that to determine the integration limits? Once I get those, I'm ok with the integration process but defining those values is where I get stuck.

Any pointers will be really appreciated. Thanks!

First do $$0\le z \le 2+x^2+\frac{y^2}{4}$$.
Next $$0\le y \le \sqrt{1-x^2}$$
Finally $$0\le x\le 1$$
$$y$$ and $$x$$ may switched for the last two steps if more convenient.
• $x^2+y^2=1$ limits both $x$ and $y$ to be at most $=1$. – herb steinberg Sep 8 '19 at 16:39