Answer
a) Conservative
b) $16$
Work Step by Step
a) WThe vector field $F(x,y)=ai+bj$ is known as conservative field throughout the domain $D$, when we have
$\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}$
$a$ and $b$ represents the first-order partial derivatives on the domain $D$.
From the given problem, we get $\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}=
2x$
Thus,we have the vector field $F$ is conservative.
For all the three paths in the given graph the end points are same , so the line integral is same for all the three paths.
b) consider $f(x,y)=x^2y+g(y) \implies f_y(x,y)=x^2+g'(y)$
and $g(y)=k$
Now,we have $f(x,y)=x^2y+k$
Thus, $\int_C F \cdot dr =f(3,2)-f(1,2)=(18+k)-(2+k)=16$
