File:Kelgestub, via. 3D-integrale, tegning.jpg

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Description	English: Linear function to be rotated around x-axis to perform a disk integration Dansk: Tegning til 3D-integrale af kelgestub
Date	19 December 2007 (original upload date)
Source	Own work
Author	Hosianna at Danish Wikipedia
Other versions	${\begin{aligned}&y=ax+b\\&y={\frac {R-r}{x}}x^{1}+rx^{0}\\&\int _{0}^{x}({\frac {R-r}{x}}x^{1}+rx^{0})^{2}dx\pi =V\\&\int _{0}^{x}({\frac {R-r}{x}}x^{1}+rx^{0})({\frac {R-r}{x}}x^{1}+rx^{0})dx\pi =V\\&\int _{0}^{x}(({\frac {R-r}{x}}x^{1})^{2}+(rx^{0})^{2}({\frac {R-r}{x}}x^{1})+(rx^{0}))dx\pi =V\\&\int _{0}^{x}(({\frac {R-r}{x}})^{2}x^{2}+r^{2}x^{0}+2({\frac {R-r}{x}}x^{1})(r))dx\pi =V\\&\int _{0}^{x}({\frac {(R-r)^{2}}{x^{2}}}x^{2}+r^{2}x^{0}+2{\frac {R-r}{x}}x^{1}r)dx\pi =V\\&\int _{0}^{x}({\frac {R^{2}+r^{2}-Rr}{x^{2}}}x^{2}+r^{2}x^{0}+2{\frac {R-r}{x}}rx^{1})dx\pi =V\\&dx=ax^{n}={\frac {1}{1+n}}ax^{1+n}\\&\int _{0}^{x}({\frac {1}{3}}{\frac {R^{2}+r^{2}-2Rr}{x^{2}}}x^{3}+r^{2}x^{1}+{\frac {1}{2}}2{\frac {R-r}{x}}rx^{2})\pi =V\\&\int _{0}^{x}({\frac {1}{3}}(R^{2}+r^{2}-2Rr)x+r^{2}x+(R-r)rx)\pi =V\\&\int _{0}^{x}({\frac {1}{3}}(R^{2}x+r^{2}x-2Rrx)+r^{2}x+(Rr-r^{2})x)\pi =V\\&\int _{0}^{x}({\frac {1}{3}}(R^{2}x+r^{2}x-2Rrx)+r^{2}x+(Rrx-r^{2}x))\pi =V\\&\int _{0}^{x}({\frac {1}{3}}R^{2}x+{\frac {1}{3}}r^{2}x-{\frac {1}{3}}2Rrx+r^{2}x+Rrx-r^{2}x)\pi =V\\&\int _{0}^{x}({\frac {1}{3}}R^{2}x+{\frac {1}{3}}r^{2}x-{\frac {2}{3}}Rrx+r^{2}x+Rrx-r^{2}x)\pi =V\\&\int _{0}^{x}({\frac {1}{3}}R^{2}x+{\frac {1}{3}}r^{2}x+{\frac {1}{3}}Rrx)\pi =V\\&\int _{0}^{x}({\frac {1}{3}}(R^{2}x+r^{2}x+Rrx))\pi =V\\&\int _{0}^{x}({\frac {1}{3}}x(R^{2}+r^{2}+Rr))\pi =V\\&\int _{0}^{x}\pi {\frac {1}{3}}x(R^{2}+r^{2}+Rr)=V\\&x=h\\&\pi *{\frac {1}{3}}h(R^{2}+r^{2}+Rr)=V\end{aligned}}$

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This work has been released into the public domain by its author, Hosianna at Danish Wikipedia. This applies worldwide.
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Hosianna grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Original upload log

The original description page was here. All following user names refer to da.wikipedia.

2007-12-19 19:58 Hosianna 262×199× (6572 bytes)
2007-12-18 21:38 Hosianna 594×483× (28002 bytes) Tegning til bevis på formel for rumfang til en keglestub via. 3D-integrale
2007-12-18 21:30 Hosianna 183×119× (3830 bytes) Tegning til bevis for formel til rumfanget på en keglestub via. 3D-integrale
2007-12-18 21:16 Hosianna 1483×1207× (89152 bytes) Tegning til 3D-integrale af kelgestub

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	Date/Time	Thumbnail	Dimensions	User	Comment
current	14:15, 28 May 2009		262 × 199 (6 KB)	MGA73bot (talk \| contribs)	{{BotMoveToCommons\|da.wikipedia\|year={{subst:CURRENTYEAR}}\|month={{subst:CURRENTMONTHNAME}}\|day={{subst:CURRENTDAY}}}} {{Information \|Description={{da\|Tegning til 3D-integrale af kelgestub}} \|Source=Transferred from [http://da.wikipedia.org da.wikipedia

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File:Kelgestub, via. 3D-integrale, tegning.jpg

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