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Mathematics-Online lexicon:

Volume of a Solid of Revolution

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The volume $ V$ of the solid, which is generated by rotation of the graph $ r=f(x)\ge0$ , $ a\le x\le b$ around the $ x$ -axis, can be calculated with integration over circular planes:

$\displaystyle V = \pi \int_a^b f(x)^2\,dx\,


Alternatively one can integrate over the nappes of cylinders.

$\displaystyle V = \pi c^2 (b-a)+ 2\pi \int_c^d r h(r)\,dr\,

where $ c$ ($ d$ ) is the minimal (maximal) radius $ r$ and $ h(r)$ is the height of the nappe of the cylinder with radius $ r$ which is contained in the solid. This variant particularly makes sense for monotone functions $ f$ of the radius.

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  automatically generated 4/ 8/2008