Arc Length - Maple Help

Student[Calculus1]

 ArcLength
 find the arc length of a curve

 Calling Sequence ArcLength(f(x), x = a..b, opts) ArcLength(f(x), a..b, opts) ArcLength([f(x), g(x)], x = a..b, opts) ArcLength([f(x), g(x)], a..b, opts)

Parameters

 f(x) - algebraic expression in variable 'x' g(x) - algebraic expression in variable 'x' x - name; specify the independent variable a, b - algebraic expressions; specify the end points of the curve opts - equation(s) of the form option=value where option is one of coordinates, functionoptions, integraloptions, integrandoptions, output, showfunction, showintegral, showintegrand, or Student plot options; specify output options

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$
 > $\mathrm{ArcLength}\left(x-3,x=-1..1\right)$
 ${2}{}\sqrt{{2}}$ (1)
 > $\mathrm{ArcLength}\left(\mathrm{ln}\left(x\right),x=1..4\right)$
 ${-}\sqrt{{2}}{+}{\mathrm{arctanh}}{}\left(\frac{\sqrt{{2}}}{{2}}\right){+}\sqrt{{17}}{-}{\mathrm{arctanh}}{}\left(\frac{\sqrt{{17}}}{{17}}\right)$ (2)
 > $\mathrm{ArcLength}\left(\left[\mathrm{cos}\left(x\right),\mathrm{sin}\left(x\right)\right],x=0..\mathrm{\pi }\right)$
 ${\mathrm{\pi }}$ (3)
 > $\mathrm{ArcLength}\left(\mathrm{cos}\left(\mathrm{\theta }\right),\mathrm{\theta }=0..2\mathrm{\pi },\mathrm{coordinates}=\mathrm{polar}\right)$
 ${2}{}{\mathrm{\pi }}$ (4)
 > $\mathrm{ArcLength}\left(\left[\mathrm{sqrt}\left(x\right),\mathrm{ln}\left(x\right)\right],x=1..10,\mathrm{coordinates}=\mathrm{polar}\right)$
 ${-}\sqrt{{5}}{+}{5}{}\sqrt{{2}}$ (5)
 > $\mathrm{ArcLength}\left({x}^{\frac{3}{2}},x=1..4,\mathrm{output}=\mathrm{integral}\right)$
 ${{\int }}_{{1}}^{{4}}\frac{\sqrt{{9}{}{x}{+}{4}}}{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (6)
 > $\mathrm{ArcLength}\left(\mathrm{cosh}\left(x\right),x=1..4,\mathrm{output}=\mathrm{integral}\right)$
 ${{\int }}_{{1}}^{{4}}\sqrt{{{\mathrm{sinh}}{}\left({x}\right)}^{{2}}{+}{1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (7)
 > $\mathrm{simplify}\left(\right)$
 ${{\int }}_{{1}}^{{4}}{\mathrm{cosh}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (8)
 > $\mathrm{ArcLength}\left(\mathrm{sin}\left({x}^{2}\right),x=0..5,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{ArcLength}\left(\mathrm{exp}\left(\mathrm{\theta }\right),\mathrm{\theta }=0..\frac{1}{2}\mathrm{\pi },\mathrm{coordinates}=\mathrm{polar},\mathrm{output}=\mathrm{plot},\mathrm{scaling}=\mathrm{constrained},\mathrm{labels}=\left[\mathrm{\theta },y\right]\right)$
 > $\mathrm{ArcLength}\left(\mathrm{sin}\left(x\right)+x,3..10,\mathrm{output}=\mathrm{plot},\mathrm{functionoptions}=\left[\mathrm{thickness}=2\right],\mathrm{integrandoptions}=\left[\mathrm{linestyle}=\mathrm{dash}\right]\right)$