Suppose we consider surface $S$ of $z = g(x,y)$ where $(x,y)$ belongs to a suitable domain $D$ then the surface area is given by the formula $$ Area = \int\int_D \sqrt{g_x^2 + g_y^2 +1} \, dA $$ where $g_x$ and $g_y$ denote partial derivatives of $g$ with respect to $x$ and $y$.
Now if we consider two surfaces, $$ z = g(x,y) \text{ and } z = h(x,y) $$ where $(x,y) \in D$ then what is the surface area of the surface formed by the intersection of $z= g(x,y)$ and $z= h(x,y)$?
Some examples :
If I consider $z = h(x,y) = c $ where $c$ is some constant (we assume that this plane intersects with $z = g(x,y)$) then life becomes easy. The area will be $$\int\int \sqrt{g_x^2 + g_y^2 +1} \, dA $$ integrated over the region $g(x,y) = c$ in $D$.
How to deal with general case? (Continuity and differentibilty of $g$ and $h$ can be assumed for simplicity)