The figure below open cylindrical can, \( S \), standing on the \( x y \)-plane. ( \( S \) has a bottom and sides, but no top.)
The side of \( S \) is given by \( x^{2}+y^{2}=16 \), and its height is 3 .
(a) Give a parametric equation, \( \vec{r}(t) \) for the rim, \( C \).
\( \vec{r}(t)=<4 \mathrm{tcostheta,4tsintheta,4t>} \)
with
0
\( \leq t \leq \)
2pi
(For this problem, enter your vector equation with angle-bracket notation: \( <f(t), g(t), h(t)> \).)
(b) If \( S \) is oriented outward and downward, find \( \int_{S} \operatorname{curl}(-y \vec{i}+x \vec{j}+4 z \vec{k}) \cdot d \vec{A} \).
\[
\int_{S} \operatorname{curl}(-y \vec{i}+x \vec{j}+4 z \vec{k}) \cdot d \vec{A}=
\]
\( \square \)
