Chapter 6.4, Problem 26ES

These exercises assume familiarity with the basic concepts of parametric curves. If needed, an introduction to this material is provided in Web Appendix $\text{M}$ .

Assume that no segment of the curve

$x=x\left(t\right),y=y\left(t\right),\left(a\le t\le b\right)$

is traced more than once as $r$ increases from $a$ to $b$ . Divide the interval $\left[a,b\right]$ into $n$ subintervals by inserting points $t_1,t_2,\dots ,t_{n-1}$ between $a=t_0$ and $b=t_n$ . Let $L$ denote the arc length of the curve. Give an informal argument for the approximation.

$L\approx {\displaystyle \underset{}{\overset{}{\sum }}\sqrt{{\left[x\left(t_k\right)-x\left(t_{k-1}\right)\right]}^2+{\left[y\left(t_k\right)-y\left(t_{k-1}\right)\right]}^2}}$

If $dx/dt$ and $dy/dt$ are continuous functions for $a\le t\le b$ , then it can be shown that as max $\Delta t_k\to 0$ , this sum converges to

$L={\displaystyle {\int }_a^b\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt}$