How to find the volume of a 3D ellipse with an elliptical void and a changing thickness?

Question

I am currently trying to find the volume of this frustum but I am facing some difficulty. I was given certain values and the way I imagined is that there is a frustum within the larger frustum since the top and bottom are slanted with a diagonal of a frustum left. There are a few unknown variables that are holding me back from finding the volume but I can't seem to find it. Is there any method that could lead to the discovery of the 4 unknowns or at the very least calculate the volume without finding these unknowns.

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Answer 1

How to find the volume of a $3\mathrm{D}$ ellipse with an elliptical void and a changing thickness?

An "$3\mathrm{D}$ ellipse" is called an "Ellipsoid". The general ellipsoid, also known as triaxial ellipsoid, is a quadratic surface which is defined in Cartesian coordinates as: $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ where ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ are the length of the semi-axes.

The intersection of the ellipsoid ${\displaystyle E_{abc}}$ with a level in height ${\displaystyle z}$ is the ellipse ${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1-{\frac {z^{ 2}}{c^{2}}}}$ with the semi-axes ${\displaystyle a'=a{\sqrt {1-{\frac {z^{2}}{c^{2}}}}},\ b'=b{\sqrt {1-{\frac {z ^{2}}{c^{2}}}}}}$.

The area of ​​this ellipse is ${\displaystyle A(z)=\pi \cdot a'\cdot b'=\pi \cdot a \cdot b \cdot (1-{\frac {z^{2}}{c^{2}}})}$.

The volume then results from $ V = \int_{-c}^{c} A(z) ~\mathrm{d}z = \pi \cdot a \cdot b \cdot \int_{-c}^{c} (1-\frac{z^2}{c^2}) ~\mathrm{d}z$.