Function Repository Resource:

ArcLengthIntegral

Source Notebook

Generate an inactive integral or sum of integrals used for computing the arc length of an expression with given bounds

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["ArcLengthIntegral"][expr,{x,a,b}]

returns an inactive integral for computing the arc length of expr for axb.

ResourceFunction["ArcLengthIntegral"][expr,{θ,a,b},"polar"]

returns an inactive integral for computing the arc length of expr for aθb in polar coordinates.

ResourceFunction["ArcLengthIntegral"][{expr1,expr2,},{t,a,b}]

returns an inactive integral for computing the arc length of the parametric curve defined by {x1[t],x2[t],}{expr1,expr2,} for atb.

ResourceFunction["ArcLengthIntegral"][eqtn,{{x,a,b},{y,c,d}}]

returns an inactive integral for computing the arc length of the curve given by implicit equation eqtn for axb and cyd.

Details and Options

ResourceFunction["ArcLengthIntegral"][expr,{x,lo,hi}] accepts a math expression, an equation of the form y==f[x] or a list of two or more math expressions in terms of x.
In ResourceFunction["ArcLengthIntegral"][{expr1,expr2,},{x,lo,hi}], the expri are expected to be math expressions, not equations.

Examples

Basic Examples (2) 

Set up the integral for computing the arc length of a sine wave:

In[1]:=
ResourceFunction[
 "ArcLengthIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][Sin[x], {x, 0, Pi}]
Out[1]=

Activate to compute the full result:

In[2]:=
Activate[%]
Out[2]=

Set up the integral that computes the arc length of a circle:

In[3]:=
ResourceFunction[
 "ArcLengthIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][y^2 + x^2 == 1, {x, y}]
Out[3]=

Scope (4) 

Return the integral for the arc length of a curve given in polar coordinates:

In[4]:=
ResourceFunction[
 "ArcLengthIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][Sin[x^2], {x, 0, Pi}, "polar"]
Out[4]=

Return the integral for the arc length of a parametric curve:

In[5]:=
ResourceFunction[
 "ArcLengthIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{Sin[x], Cos[x]}, {x, 0, Pi}]
Out[5]=

Return the integral for the arc length of the unit circle that is contained within the region bounded by 0x1 and 0y1/2:

In[6]:=
ResourceFunction[
 "ArcLengthIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^2 + y^2 == 1, {{x, 0, 1}, {y, 0, 1/2}}]
Out[6]=

You can also leave x or y unbounded:

In[7]:=
ResourceFunction[
 "ArcLengthIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^2 + y^2 == 1, {{x, 0, 1}, y}]
Out[7]=

Return the integral for the arc length of an asteroid bounded by -1x1:

In[8]:=
ResourceFunction[
 "ArcLengthIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^(2/3) + y^(2/3) == 1, {{x, -1, 1}, y}]
Out[8]=

Applications (3) 

Define and display an implicit region representing two disconnected segments of the unit circle:

In[9]:=
Region@ImplicitRegion[x^2 + y^2 == 1, {x, {y, 0, 1/2}}]
Out[9]=

Set up the integrals for disconnected segments of the region:

In[10]:=
ResourceFunction[
 "ArcLengthIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^2 + y^2 == 1, {x, {y, 0, 1/2}}]
Out[10]=

Then compute the result:

In[11]:=
Activate[%]
Out[11]=

Possible Issues (1) 

ArcLengthIntegral returns the integral for computing the unsigned arc length:

In[12]:=
ResourceFunction[
 "ArcLengthIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][Sin[x], {x, Pi, 0}]
Out[12]=

Publisher

Wolfram|Alpha Math Team

Version History

  • 2.0.0 – 23 March 2023
  • 1.0.0 – 01 April 2020

Related Resources

License Information