Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 InactiveRiemannSumToIntegral

Source Notebook

Determine an integral corresponding to a Riemann sum

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["InactiveRiemannSumToIntegral"][fun,{i,1,n},n→Infinity,x]

expresses the Riemann sum of fun over i from 1 to n as an integral in the variable x.

Examples

Basic Examples (1) 

Find the integral corresponding to the Riemann sum with summand :

In[1]:=
ResourceFunction[ "InactiveRiemannSumToIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][(1 + i/n)/n, {i, n}, n -> Infinity, x]
Out[1]=

Scope (2) 

Find the integral corresponding to the Riemann sum with summand :

In[2]:=
ResourceFunction[ "InactiveRiemannSumToIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][(i/n)/n, {i, n}, n -> Infinity, x]
Out[2]=

Find the integral corresponding to the Riemann sum with summand Cos[(1+i)/n]/n:

In[3]:=
ResourceFunction[ "InactiveRiemannSumToIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][Cos[(1 + i)/n]/n, {i, n}, n -> Infinity, x]
Out[3]=

Possible Issues (1) 

If an integral cannot be found, an inactivated form of the limit is returned with a message:

In[4]:=
ResourceFunction[ "InactiveRiemannSumToIntegral", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][(i/n)/Cos[n], {i, n}, n -> Infinity, x]
Out[4]=

Publisher

Wolfram|Alpha Math Team

Version History

  • 3.0.0 – 23 March 2023
  • 2.0.0 – 08 October 2021

Related Resources

Author Notes

To view the full source code for InactiveRiemannSumToIntegral, evaluate the following:

In[1]:=
SystemOpen[ FileNameJoin[{DirectoryName[FindFile["ResourceFunctionHelpers`"]], "InactiveRiemannSumToIntegral.wl"}]]

License Information