How to Integrate with an AX? The Surprising Power of Planimeters – Visually Explained!
Today’s mission is to reveal a surprisingly simple trick that lets you use any old ax(e) to calculate the area of plane shapes. More broadly, we’ll explore planimeters — mechanical instruments that compute area by tracing a shape’s edge. Real mathematical magic in action!00:00 Intro (a mathematical fairy tale)
07:16 Cool machines (history and types of planimeters)
09:05 Green's theorem (the most common high-level calculus explanation)
13:05 Easy peasy (the easiest and at the same time pretty rigorous explanation)
20:30 Full sweep (the best but somewhat handwaivy explanation)
26:50 AX (making sense of the ax planimeter)
31:45 Cute stuff
33:26 Thank you!
Things to check out:
Probably the best exposition of the handwaivy part of this video.
Robert L. Foote & Ed Sandifer, Area Without Integration: Make Your Own Planimeter, https://persweb.wabash.edu/facstaff/footer/papers/foote.sandifer.reprint.pdf
In general, check out Robert L. Foote's website, lots of planimeter and other gold :)
Robert L. Foote, How planimeters work, https://tinyurl.com/yc53vzan
https://persweb.wabash.edu/facstaff/footer/HOME.HTM
Richard Courant textbook
Differential and Integral Calculus (Volume II)-Blackie & Son, 1936
Nice exposition of why area = arm x roll starting with tracing a rectangle and a linear planimeter.
Paul Kunkel, The planimeter, https://web.archive.org/web/20021201115325/http://www.nas.com/~kunkel/planimeter/planimeter.htm
One of my favourite serious maths popularisations. My video follows the chapter on planimeters quite closely.
John Bryant and Christopher Sangwin, How round is your circle, Princeton Uni Press, 2011.
Moment planimeters, a really really nice twist to the planimeter story.
Christopher Sangwin, Locating the centre of mass by mechanical means
https://tinyurl.com/mr2dje5
That's the nice survey of maths machines that I mentioned in this video.
Fr. A Willers, Mathematische Maschinen und Instrumente, Akademie Verlag, Berlin, 1951.
Various takes on the Green's theorem explanation of planimeter math(s).
Tanya Leise, As the Planimeter’s Wheel Turns: Planimeter Proofs for Calculus Class, https://tleise.people.amherst.edu/HomePage/LeisePlanimeter.pdf
Oliver Knill, The Planimeter and the Theorem of Green
https://people.math.harvard.edu/~knill/teaching/math21a2000/planimeter/index.html
Bill Casselman and John Eggers, The Mathematics of Surveying: Part II. The Planimeter
https://www.ams.org/publicoutreach/feature-column/fcarc-surveying-two
Stan Wagon, Mathematica in action, Springer, 2010
https://www.amazon.com/Mathematica%C2%AE-Action-Problem-Visualization-Computation/dp/0387753664
Stan Wagon, Polar planimeter Wolfram demonstration project
Some animations in this video were produced using a customised version of the code underlying this demonstration project.
https://demonstrations.wolfram.com/ThePolarPlanimeter/
Nitty-gritty analysis of the hatchet planimeter.
Theory of the hatchet planimeter, Simon Farthing
https://www.ias.ac.in/public/Volumes/sadh/008/04/0351-0359.pdf
Bicycle tyre tracks, a different angle from which to look at the hatchet planimeter.
Robert Foote, Mark Levi, and Serge Tabachnikov, Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture
An old newpaper article by Marty and me about hatchet planimeters.
Marty Ross and Burkard Polster, Hatcheting a plan. The AGE, https://www.qedcat.com/archive_cleaned/161.html
Calculus without calculus
https://en.wikipedia.org/wiki/Visual_calculus
The book I mentioned at the end.
Tom Apostol and Mamikon Mnatsakanian. New Horizons in Geometry. Mathematical Association of America, 2012
The Haff company's website
https://www.haff.de
The manual for using my analog polar planimeter.
https://www.haff.de/wp-content/uploads/2018/05/manual317_315_313.pdf
An old planimeter applet (you need to figure out a way to run it, I use CheerpJ Applet Runner. Use at your own risk! :)
https://web.archive.org/web/20161011062058/http://www.leinweb.com/snackbar/planimtr/planimtr.htm
T-shirt: Label your axes https://www.teepublic.com/t-shirt/12403182-label-your-axes-funny-math
Soundtrack: Hourglass by Yehezkel Raz and The Prophets Song by Ian Post
Enjoy!
Burkard Receive SMS online on sms24.me
TubeReader video aggregator is a website that collects and organizes online videos from the YouTube source. Video aggregation is done for different purposes, and TubeReader take different approaches to achieve their purpose.
Our try to collect videos of high quality or interest for visitors to view; the collection may be made by editors or may be based on community votes.
Another method is to base the collection on those videos most viewed, either at the aggregator site or at various popular video hosting sites.
TubeReader site exists to allow users to collect their own sets of videos, for personal use as well as for browsing and viewing by others; TubeReader can develop online communities around video sharing.
Our site allow users to create a personalized video playlist, for personal use as well as for browsing and viewing by others.
@YouTubeReaderBot allows you to subscribe to Youtube channels.
By using @YouTubeReaderBot Bot you agree with YouTube Terms of Service.
Use the @YouTubeReaderBot telegram bot to be the first to be notified when new videos are released on your favorite channels.
Look for new videos or channels and share them with your friends.
You can start using our bot from this video, subscribe now to How to Integrate with an AX? The Surprising Power of Planimeters – Visually Explained!
What is YouTube?
YouTube is a free video sharing website that makes it easy to watch online videos. You can even create and upload your own videos to share with others. Originally created in 2005, YouTube is now one of the most popular sites on the Web, with visitors watching around 6 billion hours of video every month.
