What is the difference between surface area and scalar surface integrals?

Question

What is the difference between the surface area of a paremetrized surface and the scalar surface integral of a function in $\mathbb{R}^3$? Are they not the same thing?

Answer 1

They are when integrating the constant function 1.

Edit: The surface integral of the constant function 1 over a surface S equals the surface area of S. In other words, surface area is just a special case of surface integrals. A similar thing happens for line integrals: the line integral of the constant function 1 over a curve equals the length of the curve.