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Student[Calculus1]

  

ArcLength

  

find the arc length of a curve

 

Calling Sequence

Parameters

Description

Examples

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 fx 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 fx.  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 fx. 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 fx 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

withStudent[Calculus1]:

ArcLengthx3,x=1..1

22

(1)

ArcLengthlnx,x=1..4

2+arctanh122+17arctanh11717

(2)

ArcLengthcosx,sinx,x=0..π

π

(3)

ArcLengthcosθ,θ=0..2π,coordinates=polar

2π

(4)

ArcLengthx,lnx,x=1..10,coordinates=polar

5+52

(5)

ArcLengthx32,x=1..4,output=integral

∫14129x+4ⅆx

(6)

ArcLengthcoshx,x=1..4,output=integral

∫14sinhx2+1ⅆx

(7)

simplify

∫14coshxⅆx

(8)

ArcLengthsinx2,x=0..5,output=plot

ArcLengthⅇθ,θ=0..1π2,coordinates=polar,output=plot,scaling=constrained,labels=θ,y

ArcLengthsinx+x,3..10,output=plot,functionoptions=thickness=2,integrandoptions=linestyle=dash

See Also

plot/options

plot/typesetting

Student

Student plot options

Student[Calculus1]

Student[Calculus1][ArcLengthTutor]

 


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