Estimating the area of a surface
- Art Busch
- 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.
- Choosing two small step sizes defines two additional curves that enclose a small region on the surface.
- By using the step sizes to scale the tangent vectors, we can define a parallelogram in the plane tangent to S at Q.
- The area of that parallelogram closely approximates the area of the enclosed region on the surface.
- We can calculate the area of this parallelogram using cross products: