# Volume of parabolic cylinder bound by plane

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 :(

• Do you know how to sketch what the plane and the parabolic cylinder look like? May 31 '13 at 3:03
• I do! I have the sketch right infront of me. May 31 '13 at 3:13

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 $$x|_0^1,y|_0^{1-x},z|_0^{1-x^2}$$
$$\iiint 1 \, dx \, dy \, dz$$
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).