What would this look like...?

Question

the intersection of z=x^2 (a parabolic cylinder) and x^2+y^2=1 (a cylinder)

Answer 1

Imagine a roof gutter with a parabolic cross section, height=1, width=2 (-1 to 1). Now cut a hole in the gutter's center that exactly fits a round drain pipe of radius 1. The contour of the cut edge is what you get. It reaches to the top of the gutter.

Answer 2

looking down the z axis it is a circle
looking down the y axis it is a parabola
looking down the x axis it's a parabola going the other way