Find the line integral of $F=\sqrt[\phantom{2}]{z}i-4xj+2\sqrt[\phantom{2}]{y}k,$ from $(0,0,0)$ to $(1,1,1)$ over each

of the following paths.

a$.$ The straight$-$line path $C_1$:$r(t)=ti+tj+tk,0t1$

b$.$ The curved path $C_2$:$r(t)=ti+t^{2}j+t^{4}k,0t1$

c$.$ The path $C_3{C}_4$ consisting of the line segment from $(0,0,0)$ to $(1,1,0)$

followed by the segment from $(1,1,0)$ to $(1,1,1)$

$.$ The line integral of $F$ over the straight$-$line path $C_1$ is

Type an integer or a simplified fraction.$)$