Answer
Elliptical cylinder with an $x$ radius of $2$, $y$ radius of $3$, and $z$ radius of between $0$ and $2$.
Work Step by Step
Given: $r(u,v)=2 sinu i+3cosuj+vk$; $0\leq v\leq 2$
Write the vector equation in its equivalent parametric equations:
$x=2 sinu $, $y= 3cosu $ and $z=v$
Solving the first two parametric equation yields:
$\frac{x}{2}= sinu $ and $\frac{x}{3}= cosu $
Therefore,
$\frac{x^{2}}{2^{2}}+\frac{y^{2}}{3^{2}}= sin^{2}u+cos^{2}u$
$\frac{x^{2}}{2^{2}}+\frac{y^{2}}{3^{2}}=1$
which represents as a equation of a Elliptical cylinder with an $x$ radius of $2$, $y$ radius of $3$, and $z$ radius of between $0$ and $2$.
