Cylindrical Coordinates — Question 3

Question 3 – Solution

Strategy Use $x^2+y^2=r^2$ and interpret one coordinate at a time.

Equation $$\boxed{z=r^2},\qquad r\ge 0.$$

Horizontal traces For $z=c\ge 0$, $r=\sqrt c$, a circle centered on the $z$-axis. There are no points for $c<0$.

Vertical traces Fixing $\theta$ selects a vertical half-plane through the $z$-axis; within it, $z=r^2$ is one half of an upward-opening parabola (including its vertex).

Symmetry The equation contains no $\theta$, so rotating any point about the $z$-axis preserves the surface.