(Solved): (a) Find a vector parametric equation for the ellipse that lies on the plane \( z-(3 x+5 y)=-2 \) ...

(a) Find a vector parametric equation for the ellipse that lies on the plane \( z-(3 x+5 y)=-2 \) and inside the cylinder \( x^{2}+y^{2}=64 \). \( \overrightarrow{\boldsymbol{r}}(u, v)=\quad \) for \( 0 \leq u \leq 8 \) and \( 0 \leq v \leq 2 \pi \) (b) \( \overrightarrow{d A}=\overrightarrow{\boldsymbol{r}}_{u} \times \overrightarrow{\boldsymbol{r}}_{v}= \) (c) \( d A=\|\overrightarrow{d A}\|=\left\|\overrightarrow{\boldsymbol{r}}_{u} \times \overrightarrow{\boldsymbol{r}}_{v}\right\|= \) (d) Set up and evaluate a double integral for the surface area of the ellipse. Surface area \( = \) (a) Find a vector parametric equation for the ellipse that lies on the plane \( z-(3 x+5 y)=-2 \) and inside the cylinder \( x^{2}+y^{2}=64 \) \( \overrightarrow{\boldsymbol{r}}(u, v)=\quad \) for \( 0 \leq u \leq 8 \) and \( 0 \leq v \leq 2 \pi \) (b) \( \overrightarrow{d A}=\overrightarrow{\boldsymbol{r}}_{u} \times \overrightarrow{\boldsymbol{r}}_{v}= \) (c) \( d A=\|\overrightarrow{d A}\|=\left\|\overrightarrow{\boldsymbol{r}}_{u} \times \overrightarrow{\boldsymbol{r}}_{v}\right\|= \) (d) Set up and evaluate a double integral for the surface area of the ellipse. Surface area \( = \)