Answer
$$0$$
Work Step by Step
Stoke's Theorem: Let us consider that $C$ be the closed curve which encloses surface $S$ for a vector field $F$. This also states that the line integral and the surface integral must be equal. That is,
$\oint_C F \ dx=\iint_S (\nabla \times F) \cdot n \ dS$
Here, $\nabla \times F=\begin{vmatrix} i&j&k \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ y^2 z^3 & 2xyz^3 & 3xy^2z^2 \end{vmatrix}\\=(6xyz^2-6xyz^2, 3y^2 z^2-3y^2 z^2, 2yz^3-2yz^3) \\=(0,0,0)$
and $n=(-z_x, -z_y,1) =(0,0,0)$
Thus, the surface integral can be computed as:
$\iint_S (\nabla \times F) \cdot n \ dS=\iint_S (0,0,0) \cdot (0,0,0) \ dS\\=0$
This means that the curl is the zero vector for any closed surface by a smooth curve $C$ when the integral is zero.
