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Finding volume of an Ellipsoid bounded by an inclined plane

Posted 9 years ago

Greetings to all you good folks of Mathematica!!

I'm still pretty new to Mathematica (only worked with it for about a month or so), so I would like to seek the advice of you guys regarding a geometrical problem.

Essentially, I need to find the volume of an ellipsoid that's bounded by a single inclined plane. This plane has to intersect the 'mid-point' of the outer edge of the ellipse at an angle. I know this is somewhat confusing (even to myself), but to illustrate what I mean, please have a look at the figure below:

enter image description here

Basically, the plane would be the plane of XZ rotated clockwise about the pivot point of X.

I'm hoping this explains the problem better.

From there then, I need to find the volume of the ellipsoid underneath that plane. See the 2-D representation below:

enter image description here

I've written the ellipsoid equation to obtain the volume, but I have a huge problem in trying to define that plane as an extra condition.

In[1]:= reg=ImplicitRegion[x^2/a^2+y^2/b^2+z^2/c^2<=1 && z<=0, {z, y, x}];

In[2]:= Volume[reg, Assumptions->a>0 && b>0 && c>0]

Anyone got any ideas?

Would really appreciate any advice from you guys.

Thank you.

Regards Corse

POSTED BY: Ben Corse
Posted 9 years ago

Hi Ben,

Some time ago I solved a similar problem with a rectangular rather than ellipsoidal volume. The problem was to find the volume from the gauge reading for a tilted tank. The method should work with minor modification. I attach the notebook.

Best, David

POSTED BY: David Keith
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