## Calculus 8th Edition

Elliptical cylinder with an $x$ radius of $2$, $y$ radius of $3$, and $z$ radius of between $0$ and $2$.
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 $2$.