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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!

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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.

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  • $\begingroup$ Hi, thanks for the answer. But why you chose these particular values? How did you arrive to them? $\endgroup$ – Fernando Gómez Sep 7 '19 at 23:08
  • $\begingroup$ I mean, I think I know how you chose the value of z, and y, but why x less than 1? $\endgroup$ – Fernando Gómez Sep 8 '19 at 0:14
  • $\begingroup$ $x^2+y^2=1$ limits both $x$ and $y$ to be at most $=1$. $\endgroup$ – herb steinberg Sep 8 '19 at 16:39

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