Answer
$$1$$
Work Step by Step
Divergence Theorem: It states that the flux across the boundary of $R$ is equal to the net contraction or expansion of the vector field $F$ within the region $R$.This implies that
$\int\int F \cdot n d C=\int\int\int \nabla \cdot F dR$
Here, $R$ defines the connected and region in space which is oriented by boundary $C$ and n denotes the outward flux unit normal vector on boundary $C$.
In order to compute the net outward flux of the given vector field, we will firstly compute $\nabla \cdot F$.
Now, we have:
$\nabla \cdot F=\dfrac{\partial}{\partial x}(x^2) +\dfrac{\partial}{\partial y}(2xz)+\dfrac{\partial}{\partial y}(y^2)\\=2x$
Thus, $\int\int F \cdot n d C=\int\int\int \nabla \cdot F dR\\=\iiint 2x \ d V\\=2 \int_0^1 \int_0^1 \int_0^1 x \ dx \ dy \ dz\\=2 \int_0^1 \int_0^1[\dfrac{x^2}{2}]_0^1 \ dy \ dz \\=\int_0^1 [y]_0^1 \ dz\\=[z]_0^1 \\=1-0\\=1$
