2

$$\sum_{k=1}^n \pi(\operatorname{f}(x_{k-1})+\operatorname{f}(x_{k}))\sqrt{(\Delta x_{k})^2+(f'(c_k)\Delta x_{k})^2}$$

"This sum is not the Riemann sum of any function. because the points $x_{k-1}$, $x_{k}$ and $c_k$ are not the same."

The upper part is from a book called "Thomas' Calculus: Early Transcendentals 13th Edition"

Why? In Riemann sum variables of the functions need to be the same? If they need to be the same then again why? Sorry if this question is too dumb.

Dogukan Polat
  • 23
  • Where is your question from? Tell us something so that you can get help. – xpaul Sep 16 '22 at 13:50
  • 1
    Welcome to (posting at) Math.SE! <> Yes, your interpretation (the numbers $x_{k-1}$, $x_k$, and $c_k$ are not the same, so this expression is not a Riemann sum) is consistent with the usual definition of a Riemann sum of a function $\phi$, which (in the notation of the question) has the form$$\sum_{k=1}^n \phi(c_k),\Delta x_k.$$ – Andrew D. Hwang Sep 16 '22 at 13:56
  • As I understand it, functions in Riemann sums always has the same variables, right? Otherwise the sum is not a "Riemann" sum. – Dogukan Polat Sep 16 '22 at 14:12
  • 1
    FYI, this type of situation with setting up a definite integral was often handled in older books (before the mid 1940s) by something called Duhamel's theorem. For a recent exposition, see A lost theorem of calculus by Loeb (2002). Incidentally, although everything I've read about this (much more than is in Loeb's references) says this is due to Duhamel from 1856, it actually first appeared in an 1841 book by Duhamel. – Dave L. Renfro Sep 17 '22 at 17:58
  • 1
    Check https://math.stackexchange.com/a/3081325/72031 for Duhamel principle referred in previous comment. Your doubt is genuine and +1 for that. – Paramanand Singh Sep 18 '22 at 02:15
  • 1
    Also check https://math.stackexchange.com/a/3072835/72031 for treatment for arc length without use of Duhamel principle. – Paramanand Singh Sep 18 '22 at 02:16

1 Answers1

1

To fix terminology, suppose $\phi$ is a real-valued function on a closed, real interval $[a, b]$. A partition of $[a, b]$ is a finite sequence $(x_{k})_{k=0}^{n}$ such that $x_{0} = a$, $x_{n} = b$, and $x_{k-1} < x_{k}$ for $1 \leq k \leq n$.

Fix a partition $(x_{k})_{k=0}^{n}$. For each $k$ with $1 \leq k \leq n$:

  • Define the $k$th subinterval $I_{k} = [x_{k-1}, x_{k}]$;
  • Let $\Delta x_{k} = x_{k} - x_{k-1}$ denote the length of $I_{k}$;
  • Pick a sample point $c_{k}$ such that $x_{k-1} \leq c_{k} \leq x_{k}$.

The Riemann sum associated to this data is the expression $$ \sum_{k=1}^{n} \phi(c_{k})\, \Delta x_{k}. $$

Suppose $f$ is a real-valued function on $[a, b]$, fix a partition $(x_{k})_{k=0}^{n}$ of $[a, b]$ and a set of sample points $(c_{k})_{k=1}^{n}$, and consider the expression $$ \sum_{k=1}^{n} \pi (f(x_{k-1}) + f(x_{k})) \sqrt{(\Delta x_{k})^{2} + (f'(c_{k})\, \Delta x_{k})^{2}}. $$ Thomas's claim

This sum is not the Riemann sum of any function. because the points $x_{k-1}$, $x_{k}$ and $c_{k}$ are not the same.

expresses that there exists no function $\phi$ depending only on $f$ and $f'$ such that $$ \sum_{k=1}^{n} \pi(f(x_{k-1}) + f(x_{k})) \sqrt{1 + f'(c_k)^{2}}\, \Delta x_{k} = \sum_{k=1}^{n} \phi(c_{k})\, \Delta x_{k}. $$

In case a positive example helps, the expression $$ \sum_{k=1}^{n} \pi f(c_{k}) \sqrt{1 + f'(c_k)^{2}}\, \Delta x_{k} $$ is a Riemann sum, for the function $\phi(x) = \pi f(x) \sqrt{1 + f'(x)^{2}}$.

Andrew D. Hwang
  • 78,195