Monday, November 13, 2017
AlgCalcOne: The Mystery of "Circular Area" (English)
What is the area of a circle of radius 1?..The truth is that even the definition of the "area" of a unit circle is problematic. While we can get at this in an approximate fashion as laid out for us by Archimedes, getting an exact definition is illusory.https://www.youtube.com/watch?v=VtlOaQy2DfU
The main calculation that approximates pi due to Archimedes is to approximate the "length" of the circumference of a unit circle by inscribed and circumscribed regular polygons with 6,12,24,48 and then 96 sides. [Note: In the video I mistakenly discuss areas instead of arclengths around 17:29, but the conclusion is the same anyway.]
Some key Calculus ideas will be examined in this video, as we come to appreciate the huge gulf between the exact area computation of a parabolic arc and the approximate area computation for a circular arc. This understanding represents a major departure from established orthodoxy, so hold onto your hats!
AlgCalcOne: Parabolic Splines and Archimedes (English)
This is our first serious integral calculus computation: we get the formula for the signed area of a parabolic arc, parallel to what Archimedes remarkably accomplished more than 2000 years ago, but with quite a different approach. We recover Archimedes' relation to the signed area of a maximal triangle inscribed in a parabolic arc: at least for the case of y=x^2..https://www.youtube.com/watch?v=bPvinbhxmU0
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