First of all, I tried to sketch out how it would look like. For the two cylinders intersection case I was eventually able to produce something like the picture below.
My "tube" looks like a kind of flattened out, but trust me it is round.
I hope the idea is clear.
Apparently, with increasing the number of cylinders this figure will have more and more facets and consequently supposed to turn in the sphere.
The area will consist of eight parts of so called "cylindrical hoof". I found it here
with a formula for its volume.
It is obvious, that in case with n cylinders we will have n*4 cylindrical hoofs of the same size. Therefore, according to some relations that should be clear from the picture we can find the volume of the area.
Evaluating the limit with n->infinite, we use the fact that sin(PI/n)~PI/n:
Which is indeed the area of the sphere!!