Surface intersected by line parallel to x-axis

Problem:

Consider $I=\int_S \vec{v} \cdot d \vec{S}=\int_S \vec{v} \cdot \hat{n} dS$, where S is a surface that is intersected once by an line parallel to the x-axis. Show that $$dS=\frac{|\nabla f|}{\partial f/ \partial x}dy \ dx$$, and that therefore, $$I=\int_S \vec{v} \cdot \frac{\nabla f}{\partial f/ \partial x} dy \ dz$$

Attempt at a Solution:

Not really sure where to start...