Answer
$\dfrac{31}{4}$
Work Step by Step
The 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 $D$.
Consider $f(x,y)=\dfrac{x^3}{4}+g(y)$ and $f_x(x,y)=x^3$ and $f_y(x,y)=y^3$
$\implies f_y(x,y)=y^3+g'(y)$
and $g(y)=k$
Thus, we get $f(x,y)=\dfrac{x^4}{4}+\dfrac{y^4}{4}+k$; $k$ is a constant.
Hence, $W=\int_C F \cdot dr =f(2,2)-f(1,0)=\dfrac{1}{4}(32-1)=\dfrac{31}{4}$
