We all know that the area of a circle is and the volume of a
sphere is , but what about the volumes (or hypervolumes) of
balls of higher dimension? For a fun exercise I had my
multivariable calculus class compute the volumes of various
balls using multiple integrals. The surprising results inspired
this post. First some terminology. An -dimensional hypersphere
(or -sphere) of radius is the set of points in satisfying (I’ll
place the center at the origin for simplicity). For example, a
0-sphere is the two-point set on the real number line, a
1-sphere is a circle of radius in the plane, and a 2-sphere is a
spherical shell of radius in 3-dimensional space. An
-dimensional ball (or -ball) is the region enclosed by an
-sphere: the set of points in satisfying . For example, a 1-ball
is the interval , a 2-ball is a disk in the plane, and a 3-ball
is a solid ball in 3-dimensional space. It is possible to define
“volume” in —in it is length, in it is area, in it is ordinary
volume, and in it is hypervolume. Let denote