Answer
$\dfrac{4}{e}$
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 $D$.
From the given problem, we get $\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}=-2xe^{-y}$
Thus, we have the vector field $F$ is conservative.
Consider $f(x,y)=x^2e^{-y}+g(y)$
$ \implies f_y(x,y)=-x^2e^{-y}+g'(y)$
and $g(y)=k$
This implies that $f(x,y)=x^2e^{-y}+k$
Thus, $\int_C F \cdot dr =f(2,1)-f(1,0)=(1+4e^{-1}+k)-(1+k)=\dfrac{4}{e}$
