The video begins with a result proved by Archimedes about a cylinder and a sphere of the same radius. Saying only that the radii are equal is not enough to determine the volume ratio; the cylinder’s height must also be specified.
Why is the ratio 3:2?
Let the sphere have radius r and the cylinder also have radius r. Set the cylinder’s height equal to the sphere’s diameter, or 2r. The sphere’s volume is V = 4/3πr³, while the cylinder’s volume is V = πr²(2r) = 2πr³. Therefore, the cylinder-to-sphere ratio is 2πr³ : 4/3πr³ = 3:2. Reversing the order, sphere to cylinder, gives 2:3.
The height condition 2r is essential. If the cylinder’s height changes, the volume ratio changes too.
An idea that predates modern calculus
The video says Archimedes divided shapes into smaller pieces to reason about area and volume. This kind of reasoning helps explain approximation by partial sums, but it is not the same as the modern definition of an integral. OpenStax discusses the method of exhaustion and approximating area with sums in Approximating Areas.
Keep two points separate: the 3:2 ratio follows from the shapes and conditions stated above, while using smaller pieces is a historical way of thinking that connects to the ideas of limits and integrals developed later.
Original video: Watch “Why Was Calculus Invented? | History of Calculus #1” on YouTube.