$($a$)$ Find a function $f$ such that vec$(F)=$gradf and

$($b$)$ use part $($a$)$ to evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)$ along the given curve $C$

vec$(F)(x,y)=($:$x{y}^2,x^{2}y$:$)$ and $C$ is the curve given by vec$(r)(t)=($:$t+sin(\frac{t}{2}),t+cos(\frac{t}{2})$:$)$ with $0t1$