Estimating the area of a surface

• Fixed values of the parameters u and v determine a point Q on the surface.
• Fixing u but not v creates a parametric curve through Q that lies on the surface, and fixing v but not u creates another curve on the surface through Q.
• The partial derivatives and are vectors parallel to these curves, and thus also parallel to the surface. That means they define a tangent plane to the surface at Q.
• Choosing two small step sizes defines two additional curves that enclose a small "panel" on the surface.
• By using the step sizes to scale the tangent vectors, we can define a parallelogram in the tangent plane tangent to S at Q. So long as the step sizes are small, the area of that parallelogram closely approximates the area of the enclosed panel on the surface.
• We can calculate the area of this parallelogram using cross products:



• Summing over all the "pansl" on the surface gives an estimate of the total surface ares:

• Taking the limit as Δu and Δv go to zero gives