Happy birthday to me (part 3)

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)" .