spheresegmentvolume

Volume of a (hyper)sphere, sphere cap, or any band through a sphere defined by two parallel planes

This tool computes the volume inside a sphere in n dimensions, or inside a sphere cap, or inside any part of a sphere that can be defined by two parallel slicing planes.

Compute the volume of a unit, complete sphere.
In 2-d, the "volume" is pi.

V = spheresegmentvolume([],2)
V =
3.1416

V - pi
ans =
0

In 3-d, compute the volume of an exact unit hemisphere. The volume is pi*2/3, approximately 2.0944

V = spheresegmentvolume([0,1],3)
V =
2.0944

V - 2*pi/3
ans =
-4.4409e-16

In 4-d, compute the volume inside a central band around a hyper-sphere of radius 2, with the band running from -1 to +1. Thus, the parallel planes are separated by 2 units of distance and are symmetric around the center of the sphere.

V = spheresegmentvolume([-1,1],4,2)
V =
58.967

In 50 dimensions, compute the volume inside a unit hemi-spherical cap.

V = spheresegmentvolume([0,1],50)
V =
8.6511e-14