Question $10$

Let $F$:$R^2{R}^2$ be defined as $F(x,y)=(2x{y}^3,1+3x^2{y}^2).$

$($a$)$ Show that $F$ is a conservative vector field.

$($b$)$ Hence, evaluate the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}2x{y}^{3}dx+(1+3x^2{y}^2)dy$ where $C$ is the curve starting from $(1,1)$ and ending at $(2,8)$ along $y=x^3.$