$(1$ point$)$ Use Stokes' Theorem to evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}(6$xyi$+zj+2yk)*dr$

where $C$ is the intersection of the plane $x+z=6$ and the cylinder $x^2+y^2=1$ oriented counterclockwise as viewed from above.

Since the ellipse is oriented counterclockwise as viewed from above the surface we attach is oriented $?$

curl$(6$xyi$+zj+2yk)=$

$$

The easiest surface to attach to this curve is the interior of the cylinder that lies on the plane $x+z=6.$ Using this surface in Stokes' Theorem evaluate the following.

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr={\displaystyle \underset{x_1}{\overset{x_2}{}}}{\displaystyle \underset{y_1}{\overset{y_2}{}}}$

where

$y_1=$

$y_2=$

$x_1=$

$x_2=$

Evaluate

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr=$

$$