Answer
$ a.\quad$ the interior of the circle with radius 1, centered at the origin, in the xy-plane.
$ b.\quad$ the interior of the circle with radius 1, centered at (0,0,3), in the plane $z=3.$
$ c.\quad$ solid unbounded cylindrical column of radius 1 with the z-axis as its axis.
Work Step by Step
$x^{2}+y^{2}=1$ is a cylinder of radius 1 with the z-axis as its axis.
$x^{2}+y^{2}\leq 1$ are points on or inside the cylinder.
$ a.\quad$
When the solid cylinder column intersects $z=0$ (the xy-plane), the intersection is the region:
the interior of the circle with radius 1, centered at the origin, in the xy-plane.
$ b.\quad$
$z=3$ is a plane parallel to $z=0.$
When the solid cylinder column intersects $z=3$, we have the interior of the circle with radius 1, centered at (0,0,3), in the plane $z=3.$
$ c.\quad$
With no restrictions on z, the cylindrical column is unbounded from above or below.
