Mathematics-Online lexicon: Volume of a Solid of Revolution

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	Mathematics-Online lexicon:
	Volume of a Solid of Revolution

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

The volume

of the solid, which is generated by rotation of the graph $r=f(x)\ge0$ , $a\le x\le b$ around the

-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

(

) is the minimal (maximal) radius

and

is the height of the nappe of the cylinder with radius

which is contained in the solid. This variant particularly makes sense for monotone functions

of the radius.

Examples:

automatically generated 4/ 8/2008