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Drilling 3 orthogonal holes through a cube (Part 2)

I have a solid cube with its dimensions 2 units in length. I have drill-bit of radius 1 unit.
I drill a hole through the centre of each face through to the other side.
When I have finished I am left with 8 identicle corner shapes.

I would like to see the all working for the calculation of the volume of one of these corner bits?

The actual volume was provided in part 1 of the question, see :
https://www.experts-exchange.com/questions/21095902/Drilling-3-orthogonal-holes-through-a-cube.html

I would now like to see how the formula was arrived at.

SOLUTION

PointyEars

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SOLUTION

PointyEars

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ASKER CERTIFIED SOLUTION

Talmash

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JR2003

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Thanks for all your efforts. I think this one really is a tough nut to crack.
I think this problem is a suitable candidate for a numerical method like Monte Carlo integration, testing if random points are inside or outside the volume. From the link PointyEars provided in part 1 the answer is a very simple expression so you would think it wouldn't be too difficult to solve.