Talk:Center of Gravity of Cycloid

This appears to want the proof of the center of gravity of an arc, whereas the first question posed by Pascal was "How to find the area and center of gravity of the region formed between one arch of the cycloid and the x axis," i.e. of the area/region rather than of just the arc; we should probably tweak the text to show the centers of gravity for botht the arc and the region ...
 * Yes that is waht is ment --prime mover (talk) 17:24, 31 December 2019 (EST)
 * OK, so we're just going to show the center of gravity of the region between the cycloid and the x axis. --John Coupe (talk) 18:14, 31 December 2019 (EST)

I'm expecting that the center of gravity of the arc will simply be the limit of the center of gravity of the area between two slightly differently-sized cycloids.
 * dont understand --prime mover (talk) 17:24, 31 December 2019 (EST)

To calculate the center of gravity of the area we will need a formula (so that it may be re-used elsewhere) and Definition:Center of Gravity seems like a logical location for that, e.g. define it as the Definition:First Moment of Area / Area under a curve [I can't find the page for that, but surely we must have one lurking somewhere].
 * no we dont --prime mover (talk) 17:24, 31 December 2019 (EST)
 * I've just found Definition:Darboux Integral/Geometric Interpretation which mentions that the definite integral can be interpreted as the area under the graph. --John Coupe (talk) 18:14, 31 December 2019 (EST)


 * This is all just A-level mechanics, but it was a long time ago for me and I didn't enjoy it then. It's straightforward, trivial even, but the whole area of centre of gravity and all that tedious applied maths stuff just hasn't been covered at all (probably because it's so boring). If you want to make a start on covering this area of knowledge systematically, then feel free. --prime mover (talk) 19:27, 31 December 2019 (EST)

The Definition:First Moment of Area is just a generalization of Definition:Central Moments in Metric Spaces which does at least provide a bit of background/basis for the existing Definition:Moment (Probability Theory).

In turn Definition:Central Moments in Metric Spaces will need various other definitions, e.g. Definition:Borel Sigma-Algebra, but I think most, if not all, of those dependencies already exist.
 * dont know --prime mover (talk) 17:24, 31 December 2019 (EST)


 * Please note that definitions are singular on this website --prime mover (talk) 19:28, 31 December 2019 (EST)


 * ps pls sign your posts on talk pages --prime mover (talk) 17:24, 31 December 2019 (EST)
 * my bad - I'm out of practice remembering to have to do it explicitly --John Coupe (talk) 18:14, 31 December 2019 (EST)