volume as integral

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

$$V={\int }_a^b\pi {[f(x)]}^2𝑑x,$$

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

$$V={\int }_a^{b}A(t)𝑑t\cos \omega .$$