Volume of a barrel.

[Dieter]

(03-20-2016 08:25 AM)ggauny@live.fr Wrote:  In learning in hight level mathematic, we have an exercise it is how calculate the
capacity of a barrel ?
In internet is billion solution, but never the same volume is found for the same barrel !!!! ????

Sure. The exact volume depends on the exact shape of the barrel.

(03-20-2016 08:25 AM)ggauny@live.fr Wrote:  Is people have the good formula that is realy exact ?
It is very hard because they speak "parabola, eliptical generatrice" or cos, sin etc.

The is no general formula because the volume depends on how the shape of the barrel is defined, i.e. the question of how wide the barrel is at a given height. The paper you attached assumes that the arc that defines the outline of the barrel is a parabola. In your paper this is the relation between x and y. If this can be described as a parabola (y=ax²+c), the final formula is fine.

For a general solution you can only say that the volume equals the integral that is also mentioned in your paper. This leads to Guldin's second rule: the volume is Pi times the integral of the square of the function y(x).

Long time ago all this lead to what is today known as Simpson's rule for numerical integration: Johannes Kepler defined what in German is called "Keplersche Fassregel" ("Kepler's barrel rule"). He had exactly the same problem with the (more or less) exact volume of wine barrels. ;-)

Finally, the method given in your attached PDF is another approximation method. For the given example the three methods (exact, assuming a parabola shaped barrel, Kepler's rule and the one in the PDF) return a volume of 0,7958 resp. 0,7927 and 0,7907 cubic meters.

Dieter