Begin with the simpler case of three orthogonal cylinders, all of the same radius.
The key obervation you will have to make is that, whereas, in the case
of two cylinders the intersection along a suitably chosen plane is
always a square. in the case of three cylinders it will be either a
square or a square with rounded corners. The square with rounded
corners will occur when the square formed by the two cylinders whose
axes are parallel to the plane no longer fits in the circle formed by
the third cylinder. The accompanying diagram illustrates this. The
red circle represents the crosssection of a cylinder perpendicular to
the page, while the square is formed by the intersection of the two
cylinders parallel to the page. Note that if the square is so small
that its corners are contained in the circle then the intersection of
the square and circle is ismple a square. Hence, in order to solve
this problem you will have to deal with the two cases. You may decide
to break up the integral into two parts, according to the cases.