File:Riemann Sum.gif
| Uploaded by | Berto |
|---|---|
| Upload date | 2022-08-24T08:44:46Z |
| MIME type | image/gif |
| Dimensions | 1075 × 359 px |
| File size | 4.5 MB |
Summary
| Description |
English: A "Riemann integral" is an intuitive but rigorous way to define the area below a curve by approximating it as the sum of a bunch of rectangles. As the width of the rectangles goes to zero, this sum converges to the desired area. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1562066595442315267 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
| GIF development |  This diagram was created with Mathematica. |
Mathematica 13.1 code
f[x_] := (BesselJ[0, x] + 0.5)/1.5
\[Delta]values = Table[10^x, {x, 1, Log[10, 0.05], -1./100}];
frames = Table[
sum = Table[{\[Delta]min, Total@Table[ f[j*\[Delta]min]*\[Delta]min, {j, 1, 10/\[Delta]min}]}, {\[Delta]min, Select[\[Delta]values, # > \[Delta] &]}];
Grid[{{
Plot[f[x], {x, 0, 10}, PlotStyle -> {Thick, Black} , AxesStyle -> {Black, Thick}, AxesLabel -> {"x", "y"}, LabelStyle -> {Bold, Black, FontSize -> 14}, PlotRange -> {{0, 10}, {0, 1}}, Epilog -> {Orange, Opacity[0.25], EdgeForm[Directive[Thick, Orange, Opacity[0.5]]], Table[Rectangle[{j*\[Delta], 0}, {(j + 1)*\[Delta], f[j*\[Delta]]}], {j, 0, 10/\[Delta]}]}, ImageSize -> 400]
,
ListLogLinearPlot[sum, Joined -> True, PlotStyle -> {Thick, Black}, Axes -> False, Frame -> True, FrameLabel -> {"Sampling interval size", "Total area of rectangles"}, LabelStyle -> {Bold, Black, FontSize -> 14}, PlotRange -> {{0.05, 10}, {0, 4.1}}, ImageSize -> 400]
}}]
, {\[Delta], \[Delta]values}];
ListAnimate[frames]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
 
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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