$3.$ Consider the surface integral $\backslash (\backslash$iint$\_\{$S$\}\backslash$mathbf$\{$F$\}\backslash$cdot d $\backslash$mathbf$\{$S$\}\backslash ),$ where $\backslash ($ S $\backslash )$ is the closed yurt$-$shaped, oriented outward surface shown below, and $\backslash (\backslash$mathbf$\{$F$\}=\backslash$langle $3$ x$,2$ y$,$ z$\backslash$rangle $\backslash ).$ Notice that the surface comprises three separate pieces: the circular base, the cylinder wall, and the conical top. The cylinder has radius $3$ and height $2,$ and the cone has radius $3$ and height $3.$

$($a$)$ Explain what the surface integral $\backslash (\backslash$iint$\_\{$S$\}\backslash$mathbf$\{$F$\}\backslash$cdot d $\backslash$mathbf$\{$S$\}\backslash )$ tells us about the relationship between the vector field $\backslash (\backslash$mathbf$\{$F$\}\backslash )$ and the surface $\backslash ($ S $\backslash ).$

$($b$)$ Explain steps required to evaluate this surface integral directly.

$($c$)$ Explain which of the theorems from Problem $1$ can be used to determine a different type of integral with the same value as the given surface integral.

$($d$)$ Use your selected theorem from part $($c$)$ to evaluate to set up a different type of integral with the same value as the given surface integral.

$($e$)$ Interpret the new integral geometrically to find its value without evaluating it$.$