Answer
$y^{2}+z^{2}= x$
Circular Paraboloid (axis is x-axis)
Work Step by Step
Given: $r(u,v)=u^{2}i+ucosvj+usinvk$
Write the vector equation in its equivalent parametric equations:
$x=u^{2} $, $y= ucosv $ and $z=usinv$
Solving the second and third parametric equation yields:
$y^{2}+z^{2}= u^{2} cos^{2}v + u^{2}sin^{2}v$
$y^{2}+z^{2}= u^{2} (1)$
$y^{2}+z^{2}= u^{2}$
This implies,
$y^{2}+z^{2}= x$
which represents as a equation of a Paraboloid (axis is x-axis).
