The ArcLength(f(x), x=a..b) command returns the arc length of the expression expression $f\left(x\right)$ from a to b. The ArcLength([f(x), g(x)], x=a..b) command returns the parametric arc length expressed in cartesian coordinates. By using options, you can specify that the command returns a plot or inert integral instead.

In general, a closed form formula for the arc length cannot be determined. Also, most closed form formulae contain special functions that are beyond the normal scope of a first course in single-variable calculus.

If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.

The opts argument can contain any of the Student plot options or any of the following equations that (excluding coordinates and output) set plot options.

coordinates = cartesian or polar

This option specifies whether cartesian or polar coordinates are used. By default, the expressions are represented with cartesian coordinates.

functionoptions = list

A list of options for the plot of the expression $f\left(x\right)$.  By default, the expression is plotted as a solid red line. For more information on plot options, see plot/options.

integraloptions = list

A list of options for the plot of the cumulative arc length of the expression $f\left(x\right)$. By default, the cumulative arc length is plotted as a solid green line. For more information on plot options, see plot/options.

integrandoptions = list

A list of options for the plot of the integrand. By default, the integrand is plotted as a solid blue line. For more information on plot options, see plot/options.

output = value, integral, or plot

This option controls the return value of the function.

output = value specifies that the value of the arc length is returned. Other options are ignored if output = value.  This is the default.

output = integral specifies that an inert integral with the appropriate integrand is returned. Other options are ignored if output = integral.

output = plot specifies that a plot, which shows the expression, integrand of the expression, and cumulative arc length of the expression from the point a, is displayed.

showfunction = true or false

Whether the expression $f\left(x\right)$ is plotted.  By default, the value is true.

showintegral = true or false

Whether the cumulative arc length is plotted.  By default, the value is true.

showintegrand = true or false

Whether the integrand is plotted. By default, the value is true.

caption = anything

A caption for the plot.

The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right)\:$

$\mathrm{ArcLength}\left(x-3\,x=-1..1\right)$

$2\sqrt{2}$

$\mathrm{ArcLength}\left(\ln \left(x\right)\,x=1..4\right)$

$-\sqrt{2}+\mathrm{arctanh}\left(\frac{\sqrt{2}}{2}\right)+\sqrt{17}-\mathrm{arctanh}\left(\frac{\sqrt{17}}{17}\right)$

$\mathrm{ArcLength}\left(\left[\cos \left(x\right)\,\sin \left(x\right)\right]\,x=0..\mathrm{\pi }\right)$

$\mathrm{\pi }$

$\mathrm{ArcLength}\left(\cos \left(\mathrm{\theta }\right)\,\mathrm{\theta }=0..2\mathrm{\pi }\,\mathrm{coordinates}=\mathrm{polar}\right)$

$2\mathrm{\pi }$

$\mathrm{ArcLength}\left(\left[\mathrm{sqrt}\left(x\right)\,\ln \left(x\right)\right]\,x=1..10\,\mathrm{coordinates}=\mathrm{polar}\right)$

$-\sqrt{5}+5\sqrt{2}$

$\mathrm{ArcLength}\left(x^{\frac{3}{2}}\,x=1..4\,\mathrm{output}=\mathrm{integral}\right)$

${\int }_1^4\frac{\sqrt{9x+4}}{2}\ⅆx$

$\mathrm{ArcLength}\left(\cosh \left(x\right)\,x=1..4\,\mathrm{output}=\mathrm{integral}\right)$

${\int }_1^4\sqrt{{\sinh \left(x\right)}^2+1}\ⅆx$

$\mathrm{simplify}\left(\right)$

${\int }_1^4\cosh \left(x\right)\ⅆx$

$\mathrm{ArcLength}\left(\sin \left(x^2\right)\,x=0..5\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{ArcLength}\left(\exp \left(\mathrm{\theta }\right)\,\mathrm{\theta }=0..\frac{1}{2}\mathrm{\pi }\,\mathrm{coordinates}=\mathrm{polar}\,\mathrm{output}=\mathrm{plot}\,\mathrm{scaling}=\mathrm{constrained}\,\mathrm{labels}=\left[\mathrm{\theta }\,y\right]\right)$

$\mathrm{ArcLength}\left(\sin \left(x\right)+x\,3..10\,\mathrm{output}=\mathrm{plot}\,\mathrm{functionoptions}=\left[\mathrm{thickness}=2\right]\,\mathrm{integrandoptions}=\left[\mathrm{linestyle}=\mathrm{dash}\right]\right)$