Question about the definition of the arc length of the graph of a function

Question

We defined the arc length of a function as

$$L_I(f):=\int\limits_a^b\sqrt{1+(f'(x))^2}dx$$

for $I=[a,b]$ and $f\in C^1(I)$. We arrived at this formula by approximating the graph of $f$ by a series of straight lines (see here or here). I think the length of the curve is always greater or equal to the combined length of all the straight lines as a straight line is the shortest connection between two points.

My question: Why is the length of the curve equal to $L_I(f)$. I get that it can't be smaller but why is it exactly this limit? Why can't it be a number bigger than $L_I(f)$?

The arc length is defined as the supremum of the lengths of the approximating polygonal curves. — Daniel Fischer, May 29 '20 at 19:56
@DanielFischer Yes that's how I learned it too but I don't see why that definition is the actual length of the graph of $f$. There is probably another way to define the length of a curve and you can probably proof how those definitions are equivalent. Or maybe you can find an upper bound for the length of the graph and show it's equal to $L_I(f)$. I know it's a definition but there's a reason for it. — Zacharias Zarowski, May 29 '20 at 20:24
What is "the actual length"? We need to make a definition, and the definition should have a number of properties we expect from a reasonable length — coincide with our notion of length for straight line segments, be additive … . Defining the length as we do gives these properties. — Daniel Fischer, May 29 '20 at 20:34
@ZachariasZarowski: your definition of arc length is not correct. arc is best possible approximation of straight lines. — , May 29 '20 at 20:48
@DanielFischer "the actual length" as in we know the length of a straight line and we know the combined length of those straight segments is smaller than (or equal to) the length of the graph (we know the shortest path between two points is a straight line). So why did mathematicians define the length of the graph as the limit of those combined lengths as the segmentation gets finer? Why can't it be a number greater than that limit? There is probably an intuitive explanation just like how I can explain why it should be less or equal to the limit. — Zacharias Zarowski, May 29 '20 at 20:49
You might be able to find an "upper sum" of tangent line approximations (analogous to circumscribed polygon around a circle), which intuitively exceeds the "actual length," and then show the upper and lower sums converge. I don't remember enough analysis to do that, but it might work since it does with the circle. — Ned, May 29 '20 at 23:17
The only way to estimate arc length is by adding lengths of the line segments in question. In order to make this meaningful one must define the arc-length in such a manner that it can be approximated to any degree of accuracy by adding the lengths of these line segments. Thus it is natural to use supremum here. The fact that the supremum also equals the limit of these sums is technical and requires proof. — Paramanand Singh, May 30 '20 at 16:24
You should have a look at this answer : https://math.stackexchange.com/a/3072835/72031 — Paramanand Singh, May 30 '20 at 16:29
@ParamanandSingh Thank you for linking those answers. But can you please expand on why it is natural to use the supremum? I understand why the limit equals the supremum and why the supremum equals $L_I(f)$. But why is the arc length defined as $L_I(f):=\sup, {L_{\mathcal{C}} (P) \mid P\in\mathcal{P} [a, b] }$ (from your first answer)? — Zacharias Zarowski, May 30 '20 at 16:57
Because supremum of a set is greater than or equal to members of the set and can be approximated by members of the set upto any degree of accuracy. No number other than supremum has these two properties. And these two are the key requirements for arc-length. — Paramanand Singh, May 30 '20 at 17:06
@ParamanandSingh I get that the supremum and the arc length are both greater than or equal to the combined length of the straight lines but that doesn't justify them being equal for me. Why the least upper bound? Why can't it be any other upper bound? — Zacharias Zarowski, May 30 '20 at 17:23
Because we want it to be approximated as accurately as possible by adding the length of these line segments. Perhaps you are not taking note of this requirement. — Paramanand Singh, May 30 '20 at 17:34

score 0 · Answer 1 · edited May 29 '20 at 20:54

I think this is more of a comment than a concrete answer, but I can't comment yet.

I think your question is similar to asking how the integral $\int_a^b dx f(x)$ gives the exact area under the curve in the interval (a,b). One might say, thinking of the integral as a Riemann sum, $\sum_{i = 0}^{N-1} f(x_i) \Delta x_i$, the integral should give only an approximate answer to the area. However, this is not the case, of course. We get closer and closer to the answer as we reduce the length of the intervals $\Delta x_i$'s, and in fact get the exact answer as $\Delta x_i$'s $\rightarrow 0$. $\lim\limits_{\Delta x_i \rightarrow 0} \sum_{i = 0}^{N-1} f(x_i) \Delta x_i = \int_a^b dx f(x)$, which is something I assume you are already familiar with. Now my point is that the same argument can carry over for the definition of the arc length. Even though we think about the definition in terms of small straight lines, the integral is the limit where the length of these lines goes to zero, hence giving us an exact answer.

You can construct Riemann sums that are always strictly above the curve and below the curve and show that their areas are equal in the limit case (see here). To me it makes sense to say that number is the area under the graph of the function. But the definition of the arc length uses straight lines with a combined length that is less than (or equal to) the length of the curve at any point in the limit process. I want to understand why the length of the graph can't be a number greater than that limit. — Zacharias Zarowski, May 29 '20 at 21:05

Question about the definition of the arc length of the graph of a function

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