# volume as integral

The volume of a solid of revolution (http://planetmath.org/VolumeOfSolidOfRevolution) can be obtained from

 $V\;=\;\int_{a}^{b}\pi[f(x)]^{2}\,dx,$

where the integrand is the area of the intersection disc of the solid of revolution and a plane perpendicular to the axis of revolution at a certain value of $x$.  This volume formula may be generalized to an analogous formula containing instead of the area $\pi[f(x)]^{2}$ a more general intersection area $A(t)$ obtained from a given solid by cutting it with a set of parallel planes determined by the parameter $t$ on a certain axis.  One must assume that the function$t\mapsto A(t)$  is continuous on an interval$[a,\,b]$  where $a$ and $b$ correspond to the “ends” of the solid.  If the $t$-axis forms an angle (http://planetmath.org/AngleBetweenTwoLines) $\omega$ with the normal line of those planes, then we have the volume formula of the form

 $V\;=\;\int_{a}^{b}\!A(t)\,dt\,\cos\omega.$
Title volume as integral VolumeAsIntegral 2013-03-22 17:20:44 2013-03-22 17:20:44 pahio (2872) pahio (2872) 9 pahio (2872) Topic msc 51M25 msc 51-00 Volume VolumeOfSolidOfRevolution RiemannMultipleIntegral ExampleOfRiemannTripleIntegral