6.3 Integration with respect to arc length  (Page 2/2)

Before defining the integral of a step function on a curve, we need to establish the usual consistency result, encountered in the previous cases ofintegration on intervals and integration over geometric sets, the proof of which this time we put in an exercise.

Now we can make the definition of the integral of a step function on a curve.

Let $h$ be a step function on a piecewise smooth curve $C$ of finite length $L.$ The integral, with respect to arc length of $h$ over $C$ is denoted by ${\int }_{C}h\left(s\right)ds,$ and is defined by

$${\int }_{C}h\left(s\right)ds=\sum _{j=1}^n{a}_{j}L(z_{j-1},z_j),$$

where $\{z_0,z_1,...,z_n\}$ is a partition of $C$ for which $h\left(z\right)$ is the constant $a_j$ on the portion of $C$ between $z_{j-1}$ and $z_j.$

Of course, integrable functions on $C$ with respect to arc length will be defined to be functions that are uniform limits of step functions.Again, there is the consistency issue in the definition of the integral of an integrable function.

Let $C$ be a piecewise smooth curve of finite length $L.$ A function $f$ with domain $C$ is called integrable with respect to arc length on $C$ if it is the uniform limit of step functions on $C.$

The integral with respect to arc length of an integrable function $f$ on $C$ is again denoted by ${\int }_{C}f\left(s\right)ds,$ and is defined by

$${\int }_{C}f\left(s\right)ds=lim{\int }_C{h}_n\left(s\right)ds,$$

where $\left\{h_n\right\}$ is a sequence of step functions that converges uniformly to $f$ on $C.$

In a sense, we are simply identifying the curve $C$ with the interval $[0,L]$ by means of the 1-1 parameterizing function $\gamma .$ The next theorem makes this quite plain.

Let $C$ be a piecewise smooth curve of finite length $L,$ and let $\gamma$ be a parameterization of $C$ by arc length. If $f$ is an integrable function on $C,$ then

First, if $h$ is a step function on $C,$ let $\left\{z_j\right\}$ be a partition of $C$ for which $h\left(z\right)$ is a constant $a_j$ on the portion of the curve between $z_{j-1}$ and $z_j.$ Let $\left\{t_j\right\}$ be the partition of $[0,L]$ for which $z_j=\gamma \left(t_j\right)$ for every $j.$ Note that $h\circ \gamma$ is a step function on $[0,L],$ and that $h\circ \gamma \left(t\right)=a_j$ for all $t\in (t_{j-1},t_j).$ Then,

which proves the theorem for step functions.

Finally, if $f=lim{h}_n$ is an integrable function on $C,$ then the sequence $\{h_n\circ \gamma \}$ converges uniformly to $f\circ \gamma$ on $[0,L],$ and so

where the final equality follows from [link] . Hence, [link] is proved.

Although the basic definitions of integrable and integral, with respect to arc length, are made in terms of the particular parameterization $\gamma$ of the curve, for computational purposes we need to know how to evaluate these integrals usingdifferent parameterizations. Here is the result:

Let $C$ be a piecewise smooth curve of finite length $L,$ and let $\phi :[a,b]\to C$ be a parameterization of $C.$ If $f$ is an integrable function on $C.$ Then

Write $\gamma :[0,L]\to C$ for a parameterization of $C$ by arc length. As in the proof to [link] , we write $g:[a,b]\to [0,L]$ for ${\gamma }^{-1}\circ \phi .$ Just as in that proof, we know that $g$ is a piecewise smooth function on the interval $[a,b].$ Hence, recalling that $|{\gamma }^{\text{'}}\left(t\right)|=1$ and $g^{\text{'}}\left(t\right)>0$ for all but a finite number of points, the following calculation is justified:

as desired.

The final theorem of this section sums up the properties of integrals with respect to arc length. There are no surprises here.

Let $C$ be a piecewise smooth curve of finite length $L,$ and write $I\left(C\right)$ for the set of all functions that are integrable with respect to arc length on $C.$ Then:

1.   $I\left(C\right)$ is a vector space ovr the real numbers, and
   $${\int }_C(af\left(s\right)+bg\left(s\right))ds=a{\int }_{C}f\left(s\right)ds+b{\int }_{C}g\left(s\right)ds$$
   for all $f,g\in I\left(C\right)$ and all $a,b\in R.$
2.  (Positivity) If $f\left(z\right)\ge 0$ for all $z\in C,$ then ${\int }_{C}f\left(s\right)ds\ge 0.$
3. If $f\in I\left(C\right),$ then so is $\left|f\right|,$ and $|{\int }_{C}f\left(s\right)ds|\le {\int }_C\left|f\left(s\right)\right|ds.$
4. If $f$ is the uniform limit of functions $f_n,$ each of which is in $I\left(C\right),$ then $f\in I\left(C\right)$ and ${\int }_{C}f\left(s\right)ds=lim{\int }_C{f}_n\left(s\right)ds.$
5. Let $\left\{u_n\right\}$ be a sequence of functions in $I\left(C\right),$ and suppose that for each $n$ there is a number $m_n,$ for which $|u_n\left(z\right)|\le m_n$ for all $z\in C,$ and such that the infinite series $\sum m_n$ converges. Then the infinite series $\sum u_n$ converges uniformly to an integrable function, and ${\int }_C\sum u_n\left(s\right)ds=\sum {\int }_C{u}_n\left(s\right)ds.$

REMARK Because of the result in part (b) of the preceding exercise, we speak of “integrating over $C$ ” when we are integrating with respect to arc length. We do not speak of “integrating from $z_1$ to $z_2,$ ” since the direction doesn't matter. This is in marked contrast to the next two kinds ofintegrals over curves that we will discuss.

here is one final bit of notation. Often, the curves of interest to us are graphs of real-valued functions.If $g:[a,b]\to R$ is a piecewise smooth function, then its graph $C$ is a piecewise smooth curve, and we write ${\int }_{\text{graph}\left(g\right)}f\left(s\right)ds$ for the integral with respect to arc length of $f$ over $C=\text{graph}\left(g\right).$