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One can derive the volumes of various 3D shapes using volume integrals with appropriate limits.

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The volume of a cube based on the volume of cylinder?? Are you sure it wasn't the other way round?

Anyway, as Hottenanny pointed out you can just calculate these kind of volume using integrals or (essentially the same) http://fr.wikipedia.org/wiki/Principe_de_Cavalieri" [Broken] (sorry, only available in French and German)

Anyway, as Hottenanny pointed out you can just calculate these kind of volume using integrals or (essentially the same) http://fr.wikipedia.org/wiki/Principe_de_Cavalieri" [Broken] (sorry, only available in French and German)

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For simplicity consider a unit square.

We define the area of a unit square as 1. (at least I think we do!!)

Seems to say so here:-

http://mathforum.org/library/drmath/view/60392.html

I think however we could have defined the area of a unit square a 7 or 12.738 or 1/4

or even -0.0009300203.

It would just make the maths a bit harder it we did!

So it seems to me it is a definition so proving it is trivial, for example an exam question

might be:-

a) Given the area of a unit square is 1, show the area of a unit square is 1.

I don't think that will be worth too many marks!!

Going on to volume, I think we define a unit volune as one, but it could have been

defined as any number, one just makes the numbers easier to work with.

We define the area of a unit square as 1. (at least I think we do!!)

Seems to say so here:-

http://mathforum.org/library/drmath/view/60392.html

I think however we could have defined the area of a unit square a 7 or 12.738 or 1/4

or even -0.0009300203.

It would just make the maths a bit harder it we did!

So it seems to me it is a definition so proving it is trivial, for example an exam question

might be:-

a) Given the area of a unit square is 1, show the area of a unit square is 1.

I don't think that will be worth too many marks!!

Going on to volume, I think we define a unit volune as one, but it could have been

defined as any number, one just makes the numbers easier to work with.

Last edited:

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The fact that the volume of an n by n cube, for n an integer, is n

Once we are "given" the area of a circle, the proof that the volume of a right circular cylinder is [itex]\pi r^2 h[/itex] is fairly simple.

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Thanks.

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