Generalizing volumes of revolution to higher dimensions

[hy.dodec]

How can we generalize volumes of revolution (3D) to higher dimensions?

And can we evaluate bulk or surcell volume of that shape?

1. In 3D, a sphere can be constructed by rotating a half circle by coordinate axis.
Then in 4D, can 4D hypersphere be constructed by rotating half sphere by stationary plane?

2. volume of a sphere can be calculated via disk method ( Integrating cross-sectional areas , http://tutorial.math.lamar.edu/Classes/ ... Rings.aspx )
Can disk method be generalized to higher dimensions?

[quickfur]

Volumese of revolution can certainly be generalized to higher dimensions.

The cubinder, for example, can be generated by revolving a 3D cube around a plane that involves the 4th dimension.

Revolving a sphere around its center produces a 4D sphere.

Revolving a sphere around a point outside its surface produces a spherindrical torus, one of the analogues of a 3D torus (there are multiple possibilities in 4D).

Revolving a cylinder around the plane formed by its axis and the 4th axis produces a spherinder. OTOH, revolving it around the plane parallel to its lids produces instead a duocylinder.

These are just some of the simplest shapes you can make. There are many, many more.