6.2 Arc length  (Page 3/3)

Now, because ${\phi }^{\text{'}}$ is uniformly continuous on the closed interval $[t_1,t_{n-1}],$ there exists a $\delta >0$ such that $|{\phi }^{\text{'}}\left(t\right)-{\phi }^{\text{'}}\left(s\right)|<ϵ$ if $|t-s|<\delta$ and $t$ and $s$ are in the interval $[t_1,t_{n-1}].$ We may assume, again by taking a finer partition if necessary, that the mesh size of $P$ is less than this $\delta .$ Then, using part (f) of [link] , we may also assume that the partition $P$ is such that

no matter what points $s_j$ in the interval $(t_{j-1},t_j)$ are chosen. So, we have the following calculation, in the middle of which we use the Mean Value Theorem on the two functions $u$ and $v.$

This implies that

If we now let $t_1$ approach $a$ and $t_{n-1}$ approach $b,$ we get

which completes the proof, since $ϵ$ is arbitrary.

We now have all the ingredients necessary to define the length of a smooth curve.

Let $C$ be a piecewise smooth curve in the plane.The length or arc length $L\equiv L\left(C\right)$ of $C$ is defined by the formula

$$L\left(C\right)=L^{\phi }=\underset{P}{sup}{L}_P^{\phi },$$

where $\phi$ is any parameterization of $C.$

If $z$ and $w$ are two points on a piecewise smooth curve $C,$ we will denote by $L(z,w)$ the arc length of the portion of the curve between $z$ and $w.$

REMARK According to [link] and [link] , we have the following formula for the length of a piecewise smooth curve:

where $\phi$ is any parameterization of $C.$

It should come as no surprise that the length of a curve $C$ from $z_1$ to $z_2$ is the same as the length of that same curve $C,$ but thought of as joining $z_2$ to $z_1.$ Nevertheless, let us make the calculation to verify this. If $\phi :[a,b]\to C$ is a parameterization of this curve from $z_1$ to $z_2,$ then we have seen in part (f) of exercise 6.1 that $\psi :[a,b]\to C,$ defined by $\psi \left(t\right)=\phi (a+b-t),$ is a parameterization of $C$ from $z_2$ to $z_1.$ We just need to check that the two integrals giving the lengths are equal. Thus,

where the last equality follows by changing variables, i.e., setting $t=a+b-s.$

We can now derive the formula for the circumference of a circle, which was one of our main goals. TRUMPETS?

Let $C$ be a circle of radius $r$ in the plane. Then the length of $C$ is $2\pi r.$

Let the center of the circle be denoted by $(h,k).$ We can parameterize the top half of the circle by the function $\phi$ on the interval $[0,\pi ]$ by $\phi \left(t\right)=h+rcos\left(t\right)+i(k+rsin(t\left)\right).$ So, the length of this half circle is given by

The same kind of calculation would show that the lower half of the circle has length $\pi r,$ and hence the total length is $2\pi r.$

The integral formula for the length of a curve is frequently not much help, especially if you really want to know how long a curve is.The integrals that show up are frequently not easy to work out.