Section 15.5 Arc Length
We can use the
ArcLength() command to find the arc length of a function over a specified interval. To use the this command, we must load the Student[Calculus1] package. Packages are loaded using the with() command and it is typically a good idea to load any necessary packages at the top of your worksheet. Loading a package only needs to be done once per Maple worksheet, but needs to be run each time you open a new or previously closed document.
We can calculate the arclength for a simple function such as \(f(x)=x^2\) over an interval \([0,\pi]\) using a single
ArcLength() command. It can be useful to assign the function first.
> f(x) := x^2;
\begin{equation*}
\displaystyle f\, := \,x\mapsto {x}^{2}
\end{equation*}
The
ArcLength() command will attempt to give the exact value of the arclength. evalf(%) may be used to display the result as a decimal.
> with(Student[Calculus1]): > ArcLength(f(x), x=0..Pi); evalf(%)
\begin{equation*}
\displaystyle 1/2\,\pi \, \sqrt{4\,{\pi }^{2}+1}-1/4\,\ln \left( -2\,\pi + \sqrt{4\,{\pi }^{2}+1} \right)
\end{equation*}
\begin{equation*}
\displaystyle 10.62814707
\end{equation*}
In some cases, the exact value of arclength may not be able to be expressed algebraically, such as in the case of the arclength of \(g(x) = \sin(x)\) over the interval \([0,\pi]\text{.}\)
> g(x) := sin(x);
\begin{equation*}
\displaystyle g\, := \,x\mapsto \sin \left( x \right)
\end{equation*}
> arclen := ArcLength(g(x), x=0..Pi);
\begin{equation*}
\displaystyle arclen\, := \,2\, \sqrt{2}\, \mathrm{EllipticE} \left( 1/2\, \sqrt{2} \right)
\end{equation*}
\(\mathrm{EllipticE}()\) is a transcendental function that you don’t have to know too much about. However, this is a numerical value that can be written as a decimal.
> evalf(arclen);
\begin{equation*}
\displaystyle 3.820197788
\end{equation*}
To have a better understanding of why the exact value may not be able to be given algebraically, it can be useful to see what integral is involved to compute the arclength. To do this, we can add the parameter
output = integral to the ArcLength() command.
> ArcLength(g(x), x=0..Pi, output=integral);
\begin{equation*}
\displaystyle \int _{0}^{\pi }\! \sqrt{ \left( \cos \left( x \right) \right) ^{2}+1}\,\,{dx}
\end{equation*}
