Answer
a) Conservative.
We can see from the graph that in all the three paths, the end points are same. Thus, the line integral is the same for all the three paths.
(b) $16$
Work Step by Step
a) When the vector field $F(x,y)=Pi+Qj$ is a conservative field, then throughout the domain $D$, we have
$\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$
$P$ and $Q$ are the first-order partial derivatives on the domain $D$.
Here, we have $\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}=2x$
This implies that the vector field $F$ is conservative.
We can see from the graph that in all the three paths, the end points are same. Thus, the line integral is the same for all the three paths.
(b) Now, $f(x,y)=x^2y+g(y)$
$f_y(x,y)=x^2+g'(y)$
Here, we have $g(y)=C$
$C$ is a constant.
Thus, we get $f(x,y)=x^2y+C$
Now, we have $\int_C F \cdot dr =(18+C)-(2+C)=16$
