3 methods of deriving torus volume using integration

Name: 3 methods of deriving torus volume using integration
Uploaded: 2026-07-19T12:39:28+00:00
Duration: 1 min 15 s
Channel: Conor Cosnett
Description: In this instructional segment, Conor Cosnett addresses his colleague Yves and reviews three distinct calculus approaches for deriving the volume formula…

19 Jul 2026 · 1 min

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1:15

In this instructional segment, Conor Cosnett addresses his colleague Yves and reviews three distinct calculus approaches for deriving the volume formula of a torus. He first revisits the cylindrical shells method—subtracting two cylinders to form an infinitesimal shell and sweeping it outward—referencing a method Yves mentioned in a prior interview. He then introduces a polar‑coordinate disk method, where a circular disk is rotated around an axis to generate the torus volume. Finally, he outlines a Cartesian‑coordinate washer method, which sweeps a washer vertically through the torus. Throughout, Conor uses a conversational, enthusiastic tone, emphasizing the elegance and variety of these techniques.

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So how are you keeping, Yves? So Yves, I had a look into different methods of integrating the volume of a torus, i.e., deriving this formula, and I believe the method you mentioned in my interview, yes, on Friday, was the method of cylindrical shells, where you subtract two cylinders to get an infinitesimal shell And then you can sweep or integrate this shell across the torus like this, outward. Now curiously, there's many other methods of deriving the volume of a torus. A nice method is actually to get a disk— just let me execute this code again— a nice method is to start off with a disk and sweep the disk around an axis. And this can be very easily done in polar coordinate system. Yeah, so there we go. And then a simple method for Cartesian coordinate system is the washer method, which you basically sweep a washer up and down the torus, which also is very cool.