How to Calculate the Volume of a Sphere with an Off-Center Hole?

[gwiz]

Recently, I was calculating the volume of a sphere with a hole through the center. I started thinking, what if the hole was off-center by some distance e. I have searched high and low and can not find anything on this situation. I have tried calculating, but after the second integral, I can not complete the third (due to complexities of the equation).

I have found the special case of Viviani's Curve, whereby e is equal to half the hole radius, and the hole radius is half the sphere radius, but that does not help.

I know this can be solved numerically (or through a CAD program), but I'm determined to try and find an analytical solution. Has anybody come across information that may help? I can post some equations if necessary.

 

[Muphrid]

Why not just integrate the sphere like there's no hole and then integrate just over the volume of the whole, subtracting out whatever density you put there in the first place?