Consider a solid bounded by a surface formed by the intersection of seven cylinders of radius $R$. The axes of these cylinders all intersect at a single point; four of the axes are directed along the long diagonals of a cube, while the remaining three are perpendicular to the faces of the cube
Look at the image.

Hi. Maybe something like this



VolumeOfIntersection.sm (17.39 KiB) downloaded 3 time(s).

VolumeOfIntersection.pdf (442.78 KiB) downloaded 3 time(s).

Best regards
Alvao

Alvaro, thank you. Unfortunately, your calculation method results in a 4.5 percent error relative to the true volume of the object.
(The layer thickness turned out to be a bit too small.

Alvaro, the discussion concerned obtaining an exact formula. Look up "Steinmetz solid" on Wikipedia. Your approximate formula does not account for the volume of the parts cut off from the solid.
See the image

Hi Victor. The procedure looks good enough to me for the others cases.



Steinmetz Solid.pdf (2.79 MiB) downloaded 1 time(s).

Steinmetz Solid.sm (25.39 KiB) downloaded 2 time(s).

Best regards
Alvaro.