Work vs Volume: Trig Integral Relation

[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?

 

[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.

 

[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.