I've been trying to figure out how to calculate the exact area of a ruled surface. I think I've come to a solution that works, but I'm not sure it's totally accurate. The procedure is as follows: Consider two curves. A ruled surface is constructed by connecting each point on one curve to a corresponding point on the other curve. Now, taking a point on curve 1 (a), the corresponding point on curve 2 (b), and another point on curve 1 arbitrarily close to the first point (c), we form a triangle. Also, by taking another point on curve 2 close to point b (d), we can form triangle bcd. Adding the areas of abc and bcd, we have one element of area. Integrating these areas, you get the whole area. The question is, is this exact, or just an approximation?