Chapter 8.4, Problem 30E

To determine

To explain geometrically the reason of use of short horizontal line segments does not work to approximate the lengths of the curve.

Answer to Problem 30E

Using a horizontal partition does not work because the limit of the sum ${\displaystyle \sum \Delta x_k}$ as the norm of partition goes to zero will always be the length $\left(d-c\right)$ of the interval $\left(c,d\right)$ irrespective of the function which represents the curve. Similarly vertical partition also does not work.

Explanation of Solution

Given information:

The given statement is that the short horizontal line segments does not work to approximate the lengths of the curve.

Using a horizontal partition does not work because the limit of the sum ${\displaystyle \sum \Delta x_k}$ as the norm of partition goes to zero will always be the length $\left(d-c\right)$ of the interval $\left(c,d\right)$ irrespective of the function which represents the curve. Similarly vertical partition also does not work.