Chapter 8.4, Problem 31E

To determine

To explain if there is any limit to the possible length of the curve.

Answer to Problem 31E

There is no limit for the length of curve.

Explanation of Solution

Given information:

The given statement is that a curve is totally contained inside the square with the vertices $\left(0,0\right),\left(1,0\right),\left(1,1\right)\text{ and }\left(0,1\right)$ .

Consider the curve

  $y=\frac{1}{3}\sin \left(\frac{1}{x}\right)+0.5\text{ for }0<x<1$

The considered function oscillates infinitely as x approaches zero from the right, resulting in a infinite line integral, Which never terminates.

Therefore,

There is no limit for the length of curve.