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**

Not only calculus is needed, you may use some geometry;

I don't know exactly how does Brilliant rounds up your answer. Please, type it to three decimal places;

If you liked this problem, try my "Happy birthday to me (part 1)" or my "Happy birthday to me (part 2)".

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