TITLE: Integration Limits for a region bounded by 3 curves
QUESTION [0 upvotes]: I am trying to compute a triple integral to find the volume of the region bounded by the following curves,
$$
x^2 + 2y^2=2 
$$
$$
x+y+2z=2 
$$
$$
z=0
$$
How will I go about finding the integration limits for my triple integral of this bounded region?
My thinking for this problem was the following. Since I know that $z=0$, then from region $2$, I put $z=0$, and then I get that $x+y=2$ , and then solving for one of the variables, I get that $x = 2 - y$ and $y = 2 - x$. Then I can substitute this into the first equation, but then I get confused in which way I need to order the integration limits, and how to approach this problem. What are the limits for $z$ , $y$ , and $x$. Also based on the integration limits, which way will I integrate, $dxdydz$ , $dxdzdy$ , $dydxdz$ , $dydzdx$, $dzdxdy$, $dzdydx$ ?

REPLY [2 votes]: Since $x+y=2$  does not intersect the elipse $x^2+2y^2=2 $  in $Oxy$, the bounds are
$-1 \le y \le 1$ 
$-\sqrt{2-2y^2} \le x \le \sqrt{2-2y^2}$
$0 \le z \le (2-x-y)/2$
The order of integration is $dzdxdy$
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