2

I was studying how to calculate arc length from this page:

https://tutorial.math.lamar.edu/classes/calcii/arclength.aspx

And it said that

enter image description here

Why this statement $L= \lim\limits_{n \to \infty}\sum_{i=1}^{n}|P_{i-1}P_i|$ is true? Why is it exact and not approximate $L\approx \lim\limits_{n \to \infty}\sum_{i=1}^{n}|P_{i-1}P_i|$?

For example, why this $L\approx \lim\limits_{n \to \infty}\sum_{i=1}^{n}\Delta x_k + \Delta y_k$ approximation when n approaches infinity is not exactly equal to the arc length and the previous approximation is exact? Please prove it mathematically. (I don't want the intuition, thanks)

enter image description here

Elsa Hejazian
  • 177
  • Because the limit notation implies that we have reduced the segment lengths to essentially zero (and increased their number to infinity). Impossible in practice, of course, but then if you want an approximation, you should drop the limit notation – Yuriy S Dec 15 '20 at 20:42
  • In the second approximation, because of the limit notation, $\Delta x_k + \Delta y_k$ also reduced to essentially zero and increased their number to infinity, also both are impossible in practice, right? So why the second one is incorrect and the first one is correct? – Elsa Hejazian Dec 15 '20 at 20:49
  • @ Elsa Hejazian, because it's geometrically incorrect. Clearly, the sum of katets will not in general be equal to the hypotenuse. No matter how small they are. You should use the Pythagoras theorem – Yuriy S Dec 15 '20 at 20:54
  • 1
    $|P_{i-1}P_i|=\sqrt{(\Delta x_i)^2+(\Delta y_i)^2}\ne \Delta x_i+\Delta y_i$ – Piquito Dec 22 '20 at 17:35

0 Answers0