Classical Calculus struggles with Archimedes' parabolic area formula | AC and dCB Curves 4 |Wild Egg

The very first calculation in calculus was not done by Leibniz or Newton, but rather by Archimedes, when he found a remarkable formula to calculate areas cut off from a parabola by a chord. This was a very different situation from what he realized was happening with the "area" of a circle -- which ultimately can only be approximately defined and calculated -- for him with a sequence of inscribed and circumscribed polygons.

Here we look at the general statement of Archimedes' parabolic area formula, and explain why it is that this key foundational result in the subject is rarely treated in first year calculus texts: the reason is that classical calculus actually struggles to state and prove this result correctly. But as we will see the Algebraic Calculus has no such problem!

The full Algebraic Calculus course, is available at https://www.openlearning.com/courses/algebraic-calculus-one/homepage/?cl=1 while the screenshot pdf (251 pages) of all the lectures is at https://www.wildegg.com/store/p123/Algebraic_Calculus_One_screenshot_pdf.html

This is part of a series of video lectures designed for motivated amateur, or professional, mathematicians wanting to delve into pure maths research without the abstraction and complexity that currently besets much of the subject. We will aim to always connect our abstract musings to computational reality by focusing on examples and explicit, concrete calculations --- naturally without any pretense of being "able to do an infinite number of things"!

To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at https://www.youtube.com/channel/UCriFv3G22iOUidUhkIGXuhw, or become a Patron at https://www.patreon.com/njwildberger . There are already dozens of videos there, and many more will appear over the next months and years. One advantage: you get to access pdf summaries of some of the videos along with GeoGebra worksheets.

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are the Wild Egg Maths Playlists:

Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Видео Classical Calculus struggles with Archimedes' parabolic area formula | AC and dCB Curves 4 |Wild Egg канала Insights into Mathematics