Area of Surfaces of Revolution and Riemann sum $$\sum_{k=1}^n \pi(\operatorname{f}(x_{k-1})+\operatorname{f}(x_{k}))\sqrt{(\Delta x_{k})^2+(f'(c_k)\Delta x_{k})^2}$$
"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.
 A: 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 $\phi(x) = \pi f(x) \sqrt{1 + f'(x)^{2}}$.
