a) \(\displaystyle{x}={u}{\cos{{v}}},{y}={u}{\sin{{v}}},{z}={4}-{u}^{{2}}{{\cos}^{{2}}{v}}-{u}^{{2}}{\sin{{v}}}\)

\(\displaystyle{z}={4}-{u}^{{2}}\)

\(\displaystyle{r}{\left({u},{v}\right)}={<}{u}{\cos{{v}}},{u}{\sin{{v}}},{4}-{u}^{{2}}{>}\)

The domain of \(\displaystyle{0}\le{u}\le{2}\)

\(\displaystyle{0}\le{v}\le{2}\pi\)

b) \(\displaystyle{r}_{{u}}={<}{\cos{{v}}},{\sin{{v}}},-{2}{u}{>}\)

\(\displaystyle{r}_{{v}}={<}-{u}{\sin{{v}}},{u}{\cos{{v}}},{0}{>}\)

\(\displaystyle{r}{u}{x}{r}{v}={\left[\begin{array}{ccc} {i}&{j}&{k}\\{\cos{{v}}}&{\sin{{v}}}&-{2}{u}\\-{u}{\sin{{v}}}&{u}{\cos{{v}}}&{0}\end{array}\right]}\)

\(\displaystyle={i}{\left({0}+{2}{u}^{{2}}{\cos{{v}}}\right)}-{j}{\left(-{2}{u}^{{2}}{\sin{{v}}}\right)}+{k}{\left({u}{{\cos}^{{2}}{v}}+{u}{{\sin}^{{2}}{v}}\right)}={<}{2}{u}^{{2}}{\cos{{v}}},{2}{u}^{{2}}{\sin{{v}}},{u}{>}\)

c) Surface Area =\(\displaystyle{\int_{{0}}^{{2}}}\pi{\int_{{0}}^{{2}}}{\left|{r}_{{u}}{x}{r}_{{v}}\right|}{d}{u}{d}{v}\)

\(\displaystyle{\left|{r}_{{u}}{x}{r}_{{v}}\right|}=\sqrt{{4}}{u}^{{2}}{{\cos}^{{2}}{v}}+{4}{u}^{{4}}{{\sin}^{{2}}{v}}+{u}^{{2}}=\sqrt{{8}}{u}^{{4}}+{u}^{{2}}\)

\(\displaystyle={u}\sqrt{{1}}+{8}{u}^{{2}}\)

A=\(\displaystyle{\int_{{0}}^{{2}}}\pi{\int_{{0}}^{{2}}}{u}\sqrt{{1}}+{8}{u}^{{2}}{d}{u}{d}{v}\)

\(\displaystyle{1}+{84}^{{2}}={f}\)

\(\displaystyle={\int_{{0}}^{{2}}}\pi{d}{v}\int\sqrt{{f}}{d}\frac{{f}}{{16}}\)

\(\displaystyle{16}{u}{d}{u}={d}{f}\)

\(\displaystyle=\frac{{1}}{{16}}\int{v}{\int_{{0}}^{{2}}}\pi{\left[\frac{{f}^{{\frac{{3}}{{2}}}}}{{\frac{{3}}{{2}}}}\right]}_{{{u}={0}}}\)

\(\displaystyle=\frac{\pi}{{8}}\cdot\frac{{2}}{{3}}{\left[\frac{{\left({1}+{8}{u}^{{2}}\right)}^{{3}}}{{2}}{\int_{{0}}^{{2}}}\right.}\)

\(\displaystyle=\frac{\pi}{{12}}{\left[\frac{{\left({33}\right)}^{{3}}}{{2}}-{1}\right]}_{{u}}\)