We want to construct a structure made as follows: imagine that two long cylindrical pillars each with radius 1 intersect at right angles and their centers also intersect. What is the volume of this intersection?
If you cut the intersection by a horizontal plane at distance z from center, the cut will be a square with side-length 2*sqrt( 1-z^2). Integrate to get volume 16/3.
Another way is to imagine the largest possible sphere inscribed at the center of intersection. The sphere should have a radius of 1. At each cut perpendicular to the z-axis, the circle from the sphere is inscribed in the square from the intersection as well, So Area of cut-circle = (Pi/4)*Area of cut-square. This is true for all z, hence Volume of sphere = (Pi/4)*Volume of Intersection, this also gives 16/3