>>...uniform convergence does not imply convergence in length or area...
Under uniform convergence, the limit of the integrals is equal to the integral of the limit. So, this works fine for areas, or are you talking about something else?
I was trying to say that convergence in area for two-dimensional surfaces in R^3 requires convergence in normals in the same way that convergence in length for one-dimensional curves in R^2 requires convergence in normals.
For area in R^2, volume in R^3, and so on, you're definitely right.
Under uniform convergence, the limit of the integrals is equal to the integral of the limit. So, this works fine for areas, or are you talking about something else?