Answer
$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$
Elliptical cylinder with $z$ values between $0$ and $2$ inclusive.
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 equations 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 an equation of an Elliptical cylinder with $z$ values between $0$ and $2$ inclusive.
