Vault

Author

Eric W. Weisstein
April 11, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SolidGeometry/Vault.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Vault.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Plot

<<Graphics`InequalityGraphics`

InequalityPlot3D[{x^2+z^2<1,y^2+z^2<1,z>0},{x},{y},{z},Ticks->None,PlotPoints->15]

With[{r=1},
Show[
Block[{$DisplayFunction=Identity},
ParametricPlot3D[#,{z,0,r},{u,-1,1},
PlotPoints->10]&/@{
{(Sqrt[(r^2)-(z^2)]),u(Sqrt[(r^2)-(z^2)]),z},
{-(Sqrt[(r^2)-(z^2)]),u(Sqrt[(r^2)-(z^2)]),z},
{u(Sqrt[(r^2)-(z^2)]),(Sqrt[(r^2)-(z^2)]),z},
{u(Sqrt[(r^2)-(z^2)]),-(Sqrt[(r^2)-(z^2)]),z}
}
],
Boxed->False,Axes->False
]]

LIveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["vault.m",
Show[{
ParametricPlot3D[{
{Sqrt[1-z^2],-u Sqrt[1-z^2],z},{-Sqrt[1-z^2],u Sqrt[1-z^2],z},
{u Sqrt[1-z^2],Sqrt[1-z^2],z},
{-u Sqrt[1-z^2],-Sqrt[1-z^2],z}
},{z,0,1},{u,-1,1},PlotPoints->10,Boxed->False,Axes->False,DisplayFunction->Identity],
Graphics3D[Polygon[{{-1,-1,0},{-1,1,0},{1,1,0},{1,-1,0}}]]
},DisplayFunction->$DisplayFunction,opts]
]

Surface Area

Solve[{x^2+z^2==1,y^2+z^2==1},{x,y}]

4Integrate[2,{z,0,1}]

Volume

Integrate[(2 Sqrt[r^2-z^2])^2,{z,0,r}]

v5

<<Calculus`Integration`

Integrate[Boole[x^2+z^2<1&&y^2+z^2<1&&z>0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]

v5.1

Integrate[Boole[x^2+z^2<r^2&&y^2+z^2<r^2&&z>0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->r>0]//Timing

Centroid

(zz=Integrate[z Boole[x^2+z^2<r^2&&y^2+z^2<r^2&&z>0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->r>0])//Timing

(zz/((8 (r^3))/3))