Question

Identify the surface with the given vector equation. r(u, v) = 2 sin(u) i + 5 cos(u) j + v k, 0 ≤ v ≤ 5 plane hyperbolic paraboloid circular paraboloid elliptic cylinder circular cylinder

Answer 1

Note that this just produces three parametric equations:

$x(u) = 2 \sin(u)$
$y(u) = 5 \cos(u)$
$z(v) = v$

In the $x-y$ plane, this is just he parametric equation for an ellipse (as a function of u). The z is simply a linear function. 

The surface is then an ellipse extruded along the z-axis. We get a elliptic cylinder.