Definition 7.7. Arc length.
We define the arc length of the curve \(C\) to be the least upper bound of (7.2) over all partitions of the interval \([a,b]\text{.}\)
Section 7.2 Arc Length
In this section we want to define arc length, that is, the length of a curve. We suppose that \(\vect\gamma(t)\text{,}\) \(t\in [a,b]\text{,}\) is a regular parametrisation of a smooth curve \(C\text{.}\) To find its length we choose a partition \(a=:t_0\lt t_1\lt t_2\lt \dots\lt t_n:=b\text{.}\) We then approximate \(C\) by the polygon with vertices \(\vect\gamma(t_{i})\) as shown in Figure 7.6.
The length of the line segment connecting \(\vect\gamma(t_{i-1})\) to \(\vect\gamma(t_i)\) is \(\|\vect\gamma(t_i)-\vect\gamma(t_{i-1})\|\text{,}\) so the total length of the polygon is
\begin{equation}
\sum_{i=1}^n\|\vect\gamma(t_i)-\vect\gamma(t_{i-1})\|\text{.}\tag{7.2}
\end{equation}
The length of the polygon will always be smaller than the length of the curve as we always take a `shortcut’ from one point to the next on the polygon. If we take a finer partition, that is, make \(\max(t_{i}-t_{i-1})\) smaller, we expect to get a closer approximation of the length of \(C\text{.}\) This motivates the following definition.
The above definition is not very handy to compute the length of \(C\text{.}\) Using that the curve is smooth we want to derive a formula to compute its length. Setting \(\Delta t_i:=t_i-t_{i-1}\) we can rewrite (7.2) as
\begin{equation*}
\sum_{i=1}^n\Bigl\|
\frac{\vect\gamma(t_{i-1}+\Delta t_i)-\vect\gamma(t_{i-1})}{\Delta t_i}
\Bigr\|\Delta t_i.\text{.}
\end{equation*}
As \(\Delta t_i\to 0\) the \(i\)-th fraction in the above expression tends to \(\vect\gamma'(t_{i-1})\text{,}\) so the above sum is very close to
\begin{equation*}
\sum_{i=1}^n \|\vect\gamma'(t_{i-1})\|\Delta t_i
\end{equation*}
if \(\max\Delta t_i\) is small. The last sum is a Riemann sum for the function \(t\mapsto\|\vect\gamma'(t)\|\text{,}\) and converges to
\begin{equation*}
\int_a^b\|\vect\gamma'(t)\|\,dt
\end{equation*}
as \(\max\Delta t_i\) tends to zero. With some effort involved to estimate the error terms in the above procedure one can show that in fact the following is true.
Proposition 7.8. Formula for arc length.
Suppose that \(\vect\gamma(t)\text{,}\) \(t\in[a,b]\) is a regular parametrisation of the smooth curve \(C\text{.}\) Then the length, \(L(C)\text{,}\) of \(C\) is given by
\begin{align*}
L(C)\amp=\int_a^b\|\vect\gamma'(t)\|\,dt\\
\amp=\int_a^b\sqrt{\bigl(\gamma_1'(t)\bigr)^2
+\bigl(\gamma_2'(t)\bigr)^2+\dots+\bigl(\gamma_N'(t)\bigr)^2}\,dt\text{.}
\end{align*}
Note that the length of a curve is independent of the orientation of the curve. To compute the length of a piecewise smooth curve we add up the lengths of all its smooth parts.
Example 7.9.
We can use the above formula to compute the circumference of a circle of radius \(R\text{.}\)
Solution.
A possible parametrisation of such a circle is given by
\begin{equation*}
\vect\gamma(t)=(R\cos t,R\sin t)\text{,}
\end{equation*}
where \(t\in[0,2\pi]\text{.}\) Hence
\begin{equation*}
\vect\gamma'(t)=(-R\sin t,R\cos t)
\end{equation*}
and thus the circumference is
\begin{equation*}
\int_0^{2\pi}\sqrt{R^2\sin^2t+R^2\cos^2 t}\,dt
=R\int_0^{2\pi} 1\,dt
=2\pi R\text{,}
\end{equation*}
which is the well known result.
Let us finally look at two special cases of the above formula. First assume that \(f\) is a continuously differentiable function on the interval \([a,b]\text{.}\) Then the function
\begin{equation*}
\vect\gamma(t):=(t,f(t))
\end{equation*}
defines a smooth curve in \(\mathbb R^2\text{,}\) namely the graph of \(f\text{.}\) It follows that \(\vect\gamma'(t):=(1,f'(t))\text{,}\) and applying Proposition 7.8 we see that the
\begin{equation*}
\text{Length of the graph of }f
=\int_a^b\sqrt{1+(f'(t))^2}\,dt\text{.}
\end{equation*}
Next we suppose that the curve is given in polar coordinates, that is, the radius as a function of the angle: \(r=f(\varphi)\text{,}\) \(\varphi\in[\alpha,\beta]\text{.}\) Then \(x=r\cos\varphi=f(\varphi)\cos\varphi\) and \(y=r\sin\varphi=f(\varphi)\sin\varphi\text{,}\) so
\begin{equation*}
\vect\gamma(\varphi)=(f(\varphi)\cos\varphi,f(\varphi)\sin\varphi),
\qquad \varphi\in[\alpha,\beta]
\end{equation*}
is a parametrisation of the curve. If \(f\) is continuously differentiable then by the product rule
\begin{equation*}
\vect\gamma'(\varphi)=(f'(\varphi)\cos\varphi-f(\varphi)\sin\varphi,
f'(\varphi)\sin\varphi+f(\varphi)\cos\varphi)\text{,}
\end{equation*}
and thus
\begin{align*}
\|\vect\gamma'(\varphi)\|^2
\amp =\bigl(f'(\varphi)\cos\varphi-f(\varphi)\sin\varphi\bigr)^2
+\bigl(f'(\varphi)\sin\varphi+f(\varphi)\cos\varphi\bigr)^2\\
\amp =\bigl(f'(\varphi)\bigr)^2(\cos^2\varphi+\sin^2\varphi)
-2f'(\varphi)f(\varphi)\cos\varphi\sin\varphi\\
\amp \qquad +2f'(\varphi)f(\varphi)\cos\varphi\sin\varphi
+\bigl(f(\varphi)\bigr)^2(\sin^2\varphi+\cos^2\varphi)\\
\amp =\bigl(f(\varphi)\bigr)^2+\bigl(f'(\varphi)\bigr)^2\text{.}
\end{align*}
Hence, applying Proposition 7.8 we see that the
\begin{equation*}
\text{Length of the curve }r=f(\varphi)\text{ is }
\int_\alpha^\beta
\sqrt{\bigl(f(\varphi)\bigr)^2+\bigl(f'(\varphi)\bigr)^2}\,d\varphi\text{.}
\end{equation*}
