Problem $8$: Let $S$ be the solid bounded by the parabolic cylinders $y=x^2,y=x^2,$ and $z=-2y^2+2.$

a$)$ On the $xy-$plane, sketch $z-$traces of the bounds on $S$ at $z=0,z=1,$ and $z=2.$

b$)$ On the $yz-$plane, sketch the $x-$traces of the bounds on $S$ for $x=0$ and $x=\frac{1}{2}.$

c$)$ On the $xz-$plane, sketch the $y-$traces of the bounds on $S$ for $y=0$ and $y=\frac{1}{2}.$

Based on your work above, is $S$ an $xy,yz,$ or $xz$ solid $($it may be more than one$)?$ Pick one of the types, and write a triple integral which computes the volume of $S.$