Two of my favourite solids are formed by intersecting cylinders that are mutually perpendicular. When two cylinders intersect, the solid common to both cylinders is called a bicylinder. When three cylinders intersect, the solid common to both cylinders is called a tricylinder. Both the bicylinder and tricylinder now belong to a larger set of shapes called Steinmetz solids.
I wrote several articles on my wiki about these solids and how to determine their respective volumes.
Volume of the bicylinder:
Volume of the tricylinder:
Curiously, the bicylinder was used to figure out the volume of a sphere in ancient China. It was conceived by Liu Hui (c. 225 – 295 AD) during the Three Kingdoms Era. The volume was found by Zu Geng during the 5th century AD, the son of Zu Chongzhi (429 – 500 AD).
Here’s my article on how Zu Geng probably determined the solid (without integral calculus).