Happy birthday to me (part 3)

Calculus Level pending

In my birthday party, we'll use some conic hats. Some kids were invited and they did the mess you can see in the figure.

If the equations of the conic hats are:

\({c_1}(x,y)=4- \sqrt {4x²+4y²-8x-8y+8}\) and

\({c_2}(x,y)=4- \sqrt {4x²+4y²+8x+8y+8}\),

what is the volume \(V\) of their intersection, for \({c_1}(x,y) \geq 0\) and \({c_2}(x,y) \geq 0\)?

Details and assumptions


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