$(4)$ Consider the line integral $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}zdx+xdy+ydz,$ where $C$ is the ellipse obtained by intersecting the cylinder $x^2+y^2=4$ with the plane $y+z=6,$ oriented counterclockwise when viewed from above. Evaluate this line integral in two ways:

$($a$)$ By parameterizing $C$ by $x=2cost,y=2sint,z=6-2sint,$ for $0t2.$

$($b$)$ By using Stokes' Theorem. Note that $C$ is the positively oriented boundary of the part of the plane $y+z=6$ inside the cylinder $x^2+y^2=4,$ oriented so that the normals $n$ point upward.