volume as integral

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


where the integrand is the area of the intersection disc of the solid of revolution and a plane perpendicularPlanetmathPlanetmathPlanetmath 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 π[f(x)]2 a more general intersection area A(t) obtained from a given solid by cutting it with a set of parallel planesMathworldPlanetmath determined by the parameter t on a certain axis.  One must assume that the functionMathworldPlanetmathtA(t)  is continuous on an intervalMathworldPlanetmath[a,b]  where a and b correspond to the “ends” of the solid.  If the t-axis forms an angle (http://planetmath.org/AngleBetweenTwoLines) ω with the normal line of those planes, then we have the volume formula of the form

Title volume as integral
Canonical name VolumeAsIntegral
Date of creation 2013-03-22 17:20:44
Last modified on 2013-03-22 17:20:44
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Topic
Classification msc 51M25
Classification msc 51-00
Related topic Volume
Related topic VolumeOfSolidOfRevolution
Related topic RiemannMultipleIntegral
Related topic ExampleOfRiemannTripleIntegral