A few years ago I challenged my class to use the method of volumes by slicing to(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Work versus volume

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