The cross$-$section below is subject to an internal shear load of V$\_($x$)$hat$($i$)+$V$\_($y$)$hat$($j$).$ Determine the shear flow caused by this internal shear load at the midpoint of Leg CD$.$ Leave your expression in terms of Y$\_($x$)$ and H$\_($y$).$ The centroid $($Z$)$ is located $18$mm to the right and $17$mm above Node D$.$ Simplify your answer as much as possible.

t$\_($AC$)=$t$\_($BD$)=2$mm

t$\_($CD$)=$t$\_($DE$)=$t$\_($EC$)=1$mm

L$\_($AC$)=$L$\_($BD$)=20$mm

L$\_($CD$)=50$mm

L$\_($DE$)=30$mm

L$\_($EC$)=40$mm

$\backslash [($V$\_$I$)\_$x $=\backslash$frac$\{$V$\_$x $\backslash$cdot I$\_\{$xx$\}-$ V$\_$y $\backslash$cdot I$\_\{$xy$\}\}\{$I$\_\{$xx$\}\backslash$cdot I$\_\{$yy$\}-$ I$\_\{$xy$\}^2\}\backslash ]$

$\backslash [($V$\_$I$)\_$y $=\backslash$frac$\{$V$\_$y $\backslash$cdot I$\_\{$yy$\}-$ V$\_$x $\backslash$cdot I$\_\{$xy$\}\}\{$I$\_\{$xx$\}\backslash$cdot I$\_\{$yy$\}-$ I$\_\{$xy$\}^2\}\backslash ]$

$\backslash [$q$($s$)=$ q$\_0-\backslash$int$\_0^$s $\backslash$left$[($V$\_$I$)\_$x $\backslash$cdot t $\backslash$cdot x$($s$^\backslash$star$)+($V$\_$I$)\_$y $\backslash$cdot t $\backslash$cdot y$($s$^\backslash$star$)\backslash$right$]\backslash ,$ ds$^\backslash$star $\backslash ]$