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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?
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?