Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
$\displaystyle \int_C e^y ~dx + 2xe^y ~dy$
$\displaystyle C$ is the square with sides x=0, x=1, y=0, and y=1.
Green's Theorem: $\displaystyle \oint_C f(x, y) \, dx + g(x, y) \, dy = \int \int_{R} \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) \, dx \, dy$ where R is the region bounded by C.
In your case, $\displaystyle f(x, y) = e^y \Rightarrow \frac{\partial f}{\partial y} = e^y$ and $\displaystyle g(x, y) = 2x e^y \Rightarrow \frac{\partial g}{\partial x} = 2 e^y$.
Therefore $\displaystyle \oint_C e^y ~dx + 2xe^y ~dy = \int_{y=0}^{1} \int_{x = 0}^{1} e^y \, dx \, dy = \int_{y=0}^{1} e^y \, dy = e - 1$.
You mean this one :
$\displaystyle \int_{y=0}^{1} \int_{x = 0}^{1} e^y \, dx \, dy$
?
Because $\displaystyle e^y$ doesn't contain any x, we can write it this way :
$\displaystyle =\underbrace{\left(\int_{y=0}^1 e^y \, dy\right)}_{=e-1} \cdot \underbrace{\left(\int_{x=0}^1 \, dx\right)}_{=1}$