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How would I go about computing the volume of an ellipsoid with spherical caps removed and a cylindrical hole through it? I'm thinking about finding dV of a cross section (the ellipsoid has a circular horizontal cross section). Would this be the best way to go? I need to find if the volume is dependent on the radius of the sphere.

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Assuming $x$ is the major axis, you can integrate the ellipsoid using a triple integral while setting your $x$ limits to be $d\leq x\leq e$, assuming the "caps" of the ellipsoid are cut off at the planes $x=d$ and $x=e$. After that you can simply subtract the volume of the cylinder that was cut out.

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