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Make a sketch of the solid in the first octant bounded by the plane

$x + y = 1$

and the parabolic cylinder

$x^{2} + z = 1$

Calculate the volume of the solid..

I have no idea where to even start :(

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  • $\begingroup$ Do you know how to sketch what the plane and the parabolic cylinder look like? $\endgroup$
    – Shuhao Cao
    May 31 '13 at 3:03
  • $\begingroup$ I do! I have the sketch right infront of me. $\endgroup$
    – BrettD
    May 31 '13 at 3:13
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As can be seen from the plot below, we have a flat triangle with $(0,0),(0,1),(1,0)$ on $xy-$plane. So the required limits here is as $$x|_0^1,y|_0^{1-x},z|_0^{1-x^2}$$ enter image description here

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  • $\begingroup$ That's pretty. What software did you use to render that? $\endgroup$
    – obataku
    May 31 '13 at 7:16
  • $\begingroup$ It is Maple 17. It is good for teaching Calculus. $\endgroup$
    – Mikasa
    May 31 '13 at 7:16
  • $\begingroup$ +1 Really nice graph! I wish I knew how to use (and owned it!) Maple! $\endgroup$
    – amWhy
    Jun 1 '13 at 0:24
  • $\begingroup$ @amWhy: Thanks my dear $\endgroup$
    – Mikasa
    Jun 1 '13 at 5:47
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You want to take a triple integral

$$\iiint 1 \, dx \, dy \, dz$$

where $X$ is your solid. This makes sense to compute a volume, since you're adding up one unit to your integral per unit value. So all you need to do is work out what the bounds on those integrals should be so that they exactly cover your solid. (You may want to change the order of the $dx$, $dy$, and $dz$ around if it helps).

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