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?