Calculating Volume of Rotated Geometry: Algorithm or Integration?

[tuoni]

Unfortunately my geometry is a little rusty: if I have an area A(x,y), and rotate it around axis z (yielding a symmetrical, circular body), what is the algorithm to calculate volume?

Or am I remembering incorrectly, and the algorithms were for very specific shapes?

E.g. a circular cylinder with a cone on top; viewed from the side it has a cross-sectional area A, and rotated around axis z you get volume.

 

[HallsofIvy]

The only "algorithm" is "Heron's formula": the volume of a solid of revolution, that is, a two-dimensional region rotated around an axis, is the area of the region times the distance the center of the region moves. That, in turn, is the area times 2 $\pi$ times the distance from the center to the axis of rotation.

Of course, for a general region, finding the center requires integration. So you might as well do it as an integral to start with.