Answer
$$6$$
Work Step by Step
Here, we have $f(x, y)=2x+e^{y^2}$ and $g (x, y)=4y^2+e^{x^2}$
This gives: $\dfrac{\partial f}{\partial x}=2$ and $\dfrac{\partial g}{\partial y}=8y$
Next, we will use the Green's Theorem to compute the vector field.
$\oint_C (f dy-g dx)=\int_R (\dfrac{\partial f}{\partial x}+\dfrac{\partial g}{\partial y}) \ dA$
or, $=\int_{0}^{1} \int_0^{1} (8y+2) \ dy \ dx$
or, $=\int_{0}^{1} (\dfrac{8y^2}{2}+2y)_0^1 \ dx$
or, $=\int_{0}^{1} 6 \ dx$
or, $=6 (1-0)$
or, $=6$
