Reciprocal_integral.svg
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Summary
Description Reciprocal integral.svg |
English:
A plot giving some geometric intuition for why the integral of 1/x is ln n, based roughly on an explanation from
"Why is the integral of 1/x equal to the natural logarithm of x?"
. The integral from 1 to 2 is equal to the integral from 2 to 4 and the integral from 4 to 8. Each region is similar to the last except that it has been scaled vertically to 50% its previous height, and scaled horizontally to 200% its previous width. Extending this, the integral from 1 to 2
k
is
k
times the integral 1 to 2, just as ln 2
k
=
k
ln 2. Source used to generate this chart is shown below.
|
Date | |
Source |
Own work
This
W3C-unspecified
diagram was created with
Mathematica
.
|
Author | User:Dcoetzee |
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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.
http://creativecommons.org/publicdomain/zero/1.0/deed.en CC0 Creative Commons Zero, Public Domain Dedication false false |
Source
All source released under CC0 waiver.
Mathematica source to generate graph (which was then saved as SVG from Mathematica):
Plot[][{{1/x, 1 < x < 2}}], Piecewise[{{1/x, 2 < x < 4}}], Piecewise[{{1/x, 4 < x < 8}}]}, {x, 0, 8.5}, PlotRange -> {0, 2.1}, Filling -> {{2 -> Axis}, {3 -> Axis}, {4 -> Axis}}, LabelStyle -> {FontSize -> 14}, AspectRatio -> 0.9, Ticks -> {Table[i, {i, 0, 8}], Table[i*0.3, {i, 0, 8}]}]
Axis fonts were converted to Liberation Serif in Inkscape, and the graphic above the plot was added in Inkscape.