Work vs Volume: Trig Integral Relation

In summary, a few years ago, the speaker challenged their class to use the method of volumes by slicing to compute the volume of a 4-ball with knowledge of the volume of a 3-ball. This led to a challenging trigonometric integral. However, during a recent lesson on work, the speaker realized that the work integral for pumping liquid from a tank is equivalent to the cylindrical shells integral for 4-dimensional volume. By solving the relatively easy integral of work, it can be determined that the volume of a 4-ball is (pi)^2 (a^4)/2. This observation may not be a standard one, as it seems to have been overlooked for the last 2000 years.
  • #1
mathwonk
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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?
 
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  • #2
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
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.
 

FAQ: Work vs Volume: Trig Integral Relation

What is the work vs volume trig integral relation?

The work vs volume trig integral relation is a mathematical expression that shows the relationship between work and volume in a system. It is often used in physics and engineering to calculate the work done on a system as it undergoes a change in volume.

How is the work vs volume trig integral relation derived?

The work vs volume trig integral relation is derived from the fundamental principle of work, which states that work is equal to the force applied on an object multiplied by the distance the object moves in the direction of the force. By integrating this expression with respect to volume, we can derive the work vs volume trig integral relation.

What are the applications of the work vs volume trig integral relation?

The work vs volume trig integral relation has many practical applications in fields such as thermodynamics, fluid mechanics, and material science. It is used to calculate the work done by a gas during a volume change, determine the work required to compress a gas, and analyze the behavior of materials under stress and strain.

How is the work vs volume trig integral relation different from other work relations?

The work vs volume trig integral relation is different from other work relations, such as the work-energy theorem or the dot product of force and displacement, because it takes into account the change in volume of a system. This makes it more suitable for analyzing processes that involve changes in volume, such as chemical reactions or phase transitions.

Can the work vs volume trig integral relation be applied to non-ideal systems?

Yes, the work vs volume trig integral relation can be applied to non-ideal systems as well. However, in such cases, additional factors such as friction, heat transfer, and other non-conservative forces may need to be considered in the calculation of work. The integral expression may also become more complex in non-ideal systems.

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