Firstly, I tried to calculate the volumes of the cones, the cylinder volume is easy (no integration needed?).
I tried the following expression:
Integrate[2 [l/R] Sqrt[x^2 + z^2], {x, 0, R}, {z, -Sqrt[R^2 - x^2], Sqrt[R^2 - x^2]}, {y, Sqrt[x^2 + z^2], l}, Reals]
Simplify[%]
But...