Consider the curve $C$ described $by$ the position vector function

vec$(r)(t)=,on$ the interval $0t4$

and the vector field

vec$(F)(x,y,z)=($:$6xy+12x{z}^2,3x^2-4y{z}^4,12x^{2}z-8y^2{z}^3$:$)$

$In$ the space below part $(iii),$

$(i)$ Draw a sketch $of$ the curve $C.$

$(ii)$ Show that vec$(F)(x,y,z)is$ a conservative vector field.

$(iii)$ Calculate the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)$