Answer
(a) Please see the figure attached
(b) ${\bf{N}}\left( {u,v} \right) = \left( {3 + \sin v} \right)\cos u{\bf{i}} + \left( {3 + \sin v} \right)\sin u{\bf{j}} - \left( {3 + \sin v} \right)\cos v{\bf{k}}$
(c) $Area\left( S \right) \approx 144.0180$
Work Step by Step
(a) We plot $S$ from several different viewpoints. The result is given in the figure attached. From this figure we see that $S$ is best described as a "bottomless vase".
(b) We have $G\left( {u,v} \right) = \left( {\left( {3 + \sin v} \right)\cos u,\left( {3 + \sin v} \right)\sin u,v} \right)$. So,
${{\bf{T}}_u} = \frac{{\partial G}}{{\partial u}} = \left( { - \left( {3 + \sin v} \right)\sin u,\left( {3 + \sin v} \right)\cos u,0} \right)$
${{\bf{T}}_v} = \frac{{\partial G}}{{\partial v}} = \left( {\cos v\cos u,\cos v\sin u,1} \right)$
${\bf{N}}\left( {u,v} \right) = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
{ - \left( {3 + \sin v} \right)\sin u}&{\left( {3 + \sin v} \right)\cos u}&0\\
{\cos v\cos u}&{\cos v\sin u}&1
\end{array}} \right|$
${\bf{N}}\left( {u,v} \right) = \left( {3 + \sin v} \right)\cos u{\bf{i}} + \left( {3 + \sin v} \right)\sin u{\bf{j}}$
$ + \left[ { - \left( {3 + \sin v} \right)\cos v{{\sin }^2}u - \left( {3 + \sin v} \right)\cos v{{\cos }^2}u} \right]{\bf{k}}$
${\bf{N}}\left( {u,v} \right) = \left( {3 + \sin v} \right)\cos u{\bf{i}} + \left( {3 + \sin v} \right)\sin u{\bf{j}} - \left( {3 + \sin v} \right)\cos v{\bf{k}}$
Using a computer algebra system, we obtain
$||{\bf{N}}\left( {u,v} \right)|| = \frac{1}{2}\sqrt {2\left( {3 + \cos 2v} \right){{\left( {3 + \sin v} \right)}^2}} $
(c) By Theorem 1, the surface area of $S$ is
$Area\left( S \right) = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} ||{\bf{N}}\left( {u,v} \right)||{\rm{d}}u{\rm{d}}v$
$Area\left( S \right) = \frac{1}{2}\mathop \smallint \limits_{u = 0}^{2\pi } \mathop \smallint \limits_{v = 0}^{2\pi } \sqrt {2\left( {3 + \cos 2v} \right){{\left( {3 + \sin v} \right)}^2}} {\rm{d}}u{\rm{d}}v$
$ = \pi \mathop \smallint \limits_{v = 0}^{2\pi } \sqrt {2\left( {3 + \cos 2v} \right){{\left( {3 + \sin v} \right)}^2}} {\rm{d}}v$
Using a computer algebra system, we evaluate the integral and obtain
$Area\left( S \right) \approx 144.0180$
