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?
[Math] the difference between surface area and scalar surface integrals
integrationmultivariable-calculussurfacesvector-spaces
Related Solutions
A scalar field on $U \subseteq \mathbb{R}^n$ is a scalar-valued function $f : U \to \mathbb{R}$.
A vector field on $U \subseteq \mathbb{R}^n$ is a vector-valued function $f : U \to \mathbb{R}^n$.
These concepts appear in multivariable calculus. They are both functions, but the terminology is helpful in keeping track of what kind of objects the values of these functions are, scalars or vectors.
Added Later: It is also worth noting that sometimes you need to consider a vector-valued function which is not a vector field. Such a function on $U \subseteq \mathbb{R}^n$ is a function $f : U \to \mathbb{R}^m$ with $m \neq n$.
Try doing what you did to find the circumference of a circle. It's like you approximated the circumference with a bunch of vertical line segments. But no matter how many you make and how small you make them, they still just have a total lenght of $4R$. They're not doing a good job of approximating the curve. To do that, you need to make the ones closer to the top and bottom slanted and therefore longer.
The total length of these segments is just $4R$, not $2\pi R$, and even using more of them at a smaller size would not change this. Try to see the next higher dimension analogy.
Now, if you used the rectangles that these segments define (by adding in horizontal segments) you do get a good approximation of the area, which does get better as the height of these segments gets smaller. The missing area omitted by such rectangles will approach zero as $n\to\infty$. But again, this is not the case with the lengths, which are a constant $4R$ regardless of $n$.
Best Answer
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.