Work versus volume

1. Sep 23, 2008

mathwonk

A few years ago I challenged my class to use the method of volumes by slicing to
compute the volume of a 4 ball,

knowing the volume of a 3-ball. This leads to a slightly challenging trig
integral, but not out of reach for a strong

calculus student. (no one did it however.)

This Fall while teaching the concept of work, I noticed the work integral for
pumping liquid from a tank,

is just the "cylindrical shells" integral for 4 dimensional volume (except for
the factor of 2 pi).

So if they have computed the relatively easy integral of work

to empty a unit radius hemispherical tank of unit density liquid as pi/4,

it follows that the volume of a 4 ball of radius a, is (2pi)(pi/4)a^4 = (pi)^2
(a^4)/2.

Is this a standard observation?

2. Sep 23, 2008

mathwonk

so as far as we know, nobody has noticed this rather obvious fact for the last 2000 years?

i am not sure actually archimedes did not know this, but it seems to escape current calc texts.

3. Sep 24, 2008

mathwonk

in case you have not done this calculation, using slices means you have to integrate an odd power of (a^2 - x^2)^(1/2) to do even dimensional ball volumes, while using shells yields an even power.