Finding volume of an Ellipsoid bounded by an inclined plane

Questions
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1

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:

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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:

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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.

In1:= reg=ImplicitRegion[x^2/a^2+y^2/b^2+z^2/c^2<=1 && z<=0,
{z, y, x}];

In2:= 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

2

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