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GROUPS:
Calculus

Find volume of two intersecting figures

Fabiano Araujo, UFMG
Posted 11 years ago

How can I calculate the volume of the intersection between a sphere xˆ2+yˆ2+zˆ2=4 and a cylinder xˆ2+yˆ2-2*x=0 Figure attached. Thanks.

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POSTED BY: Fabiano Araujo
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Erik Mahieu
Erik Mahieu
Posted 11 years ago

Use the geometric computation functions:

In[27]:= R1 = Ball[{0, 0, 0}, 2];
R2 = Cylinder[{{1, 0, -2}, {1, 0, 2}}, 1];
R = RegionIntersection[R1, R2];

In[32]:= RegionMeasure@R

Out[32]= 16/9 (-4 + 3 \[Pi])
POSTED BY: Erik Mahieu

The equation of the sphere is easy:

\[Rho] == 2

The equation of the cylinder is:

\[Rho] == 2 Cos[\[Phi]] Csc[\[Theta]]

See SphericalPlot3D for information about rho, theta, and phi. Theta and phi can change from one text to another.

Here is a plot:

SphericalPlot3D[{{2}, {2 Cos[\[Phi]] Csc[\[Theta]]}}, {\[Theta], 0, 
  Pi}, {\[Phi], 0, 2 Pi}]

Eric
POSTED BY: Eric Johnstone
POSTED BY: Fabiano Araujo
Erik Mahieu
Erik Mahieu
Posted 11 years ago
POSTED BY: Erik Mahieu

enter image description here
POSTED BY: Simon Cadrin
Erik Mahieu
Erik Mahieu
Posted 11 years ago

The above is not the correct plot of the two volumes of subject. The centre of the sphere is at {0,0,0} i.o at (0,01/2}. As can be seen from this plot: enter image description here
POSTED BY: Erik Mahieu

enter image description here
POSTED BY: Simon Cadrin
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