Summary: Math 1550 Fall 2005
Section 31 P. Achar
Extra Credit Solutions
Due: December 6, 2005
1. (3 points) A 4-sphere is made by stacking up cross-sections that are 3-spheres. Following the pattern
from 2- and 3-spheres, set up an integral for the 4-volume of a 4-sphere of radius r.
Solution: Imagine a 4-sphere of radius r centered at the origin in 4-dimensional space. If you take a
"slice" perpendicular to the x-axis, you get a 3-sphere. Moreover, the radius of that slice is
r2 - x2.
The possible values of x are from -r to r, so to compute the 4-volume by cross-sections, we set up the
3-volume of the
3-volume of a 3-sphere