Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematics-Online course: Preparatory Course Mathematics - Analysis - Integral Calculus

Calculation of Rotational Volumes

[previous page] [next page] [table of contents][page overview]
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\, . $

\includegraphics[width=.75\textwidth]{rotation_a}

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.

(temporary unavailable)
[previous page] [next page] [table of contents][page overview]
  automatically generated 1/9/2017