I am a bit confused about finding the vector area. I understand that for a simple surface, such as a polygon it is just a magnitude of a cross product of two vectors living on its plane times the normal (with a direction given by the same cross product).
What about a more complicated example: a lampshade bound by a horizontal circle of radius 4 and the other one of radius 3, vertically separated by 5 units.
How does one find a normal to the cone? Is it correct to assume that the surface area is 0 because it is closed? And do I add the area of two circles on top and bottom or subtract them?
The vector area makes sense only for plane surfaces. For example, take the cylindrical shell. I can calculate the area ($2\pi R h$), I can calculate the normal to the surface at any point, but I don't have a unique direction for it (varies from point to point).
Also, the formula with vector product is true only for parallelogram. For a triangle, you would need to divide by $2$. Any other shapes can be described as a collection of triangles.