Problem $1$: $(10$ points$)$ Let $S$ be the solid whose base is the region of the first quadrant of the $xy-$plane below $y=4-x,$ and whose cross sections perpendicular to the $y-$axis $($i$.$e$.$ coming out of the page in $3-d)$ are squares. Find the volume of $S.$

Problem $2$: $(10$ points$)$ Find the volume of the solid whose base is the region bounded by $y=x^4$ and $y=1$ and whose cross sections perpendicular to the $x-$axis $($i$.$e$.$ coming out of the page in $3-d)$ are semi$-$circles.