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Volume of bottle from cross section

Hi,

Another geometry problem!

Attachment 21170

The diagram shows the vertical cross-section of a bottle of air refresher. The three curved parts of the diagram are circular arcs of radius 4 cm, with center O. AB is the diameter of circular base, CD is the diameter of the horizontal circular cross-section through O, and EF is the diameter of the cylindrical neck of the bottle. AF and BE are perpendicular to each other.

(a) Find AB.

(b) Hence find Volume of the bottle.

For (a) the right triangle gives AB = 5.66. But for (b) there seems to be so much happening here, not sure how to proceed.

Thanks for your help.