How do I parameterize these surfaces?

[Sho Kano]

Homework Statement

Parameterize $S={ S }_{ 1 }\bigcup { { S }_{ 2 } }$, where $S_1$ is the surface with equation $x^2+y^2=4$ bounded above by the graph of $2y+z=6$ and below by the $xy$ plane. $S_2$ is the bottom disk

Homework Equations

The Attempt at a Solution

${ S }_{ 1 }=\left< 2cosu,2sinu,v \right> ,0\le u\le 2\pi ,0\le v\le 6-2y\\ { S }_{ 2 }=\left< rcosu,rsinu,0 \right> ,0\le u\le 2\pi ,0\le r\le 2$It doesn't make sense with the z boundaries because $6-2y$ isn't in terms of $u$ and $v$...would it be $6-4sinu$?

 

[BvU]

I think that's the idea. I wouldn't know how to parametrize this in one single expression, ecxept also with a $\cup$ symbol (for the disc I would then use different parameter names).

Note: The $\ 0\le v \le 6-4\sin u \$ isn't a parametrization yet (imho).

$\LaTeX$ note: use \sin and \cos, etc. Much clearer

 

[Sho Kano]

I think that's the idea. I wouldn't know how to parametrize this in one single expression, ecxept also with a $\cup$ symbol (for the disc I would then use different parameter names).

Note: The $\ 0\le v \le 6-4\sin u \$ isn't a parametrization yet (imho).

$\LaTeX$ note: use \sin and \cos, etc. Much clearer

this is OK in preparation for a surface integral right? Next I'd compose $F$ (the vector field) with $S_1$ and $S_2$ with two integrals, then dot that with the normal from each parameterization. But it's not clear which normal would I use? Inner or outer? is outer the conventional choice?

 

[BvU]

What integral ? Have we seen the full problem statement yet ?