# Estimating the area of a surface

- Author:
- 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:

- Summing over all the "cells" of the surface gives an estimate of the total surface ares:

- Taking the limit as Δu and Δv go to zero gives: