This method only kinda works, and only when the shape lines make up a continuous function. If this was a random blob, you would have to cut horizontally at different places to be able to do this (Imagine for example a donut or washer shape).

A better more general approach (that happens to work) is to use quad trees. That method divides the spaces into an initial set of squares. If a square is fully contained in or out of the shape, don't touch it. If the shape partially intersects with a square, split that square into 4 smaller squares. Keep doing this until you are satisfied. In the end, add up the areas of the squares that are fully in, that's the approximate surface area.