I live in a country where, (about two generations) in the past, industrial and agricultural child labour was deemed normal. Not quite as bad as even earlier, but if you were poor (or a girl), you didn't get an education beyond the bare minimum required to do manual work.

The state then enforced a mandatory period of seven years of public schooling, where the cost of education was socialized. A vast majority of people considered this a proud achievement. (Hard) child labour is now considered a form of abuse.

Public schooling is a compromise, mostly due to financing: you can certainly stimulate children better with a lower ratio of pupils per teacher, but not everyone has the money to pay for extremes. For this reason, there are now some private schools. Expensive, more teachers per child, arguably better results. But most children go to public schools.

Home schooling is very rare, because most parents don't have the time to do it, as they have (or choose) to work. When it is done, it is met with some suspicion regarding the motivation of the parents. There have been cases of cults and religious child abuse.

Exceptions are, for instance, families who travel extensively (sailing around the world, carnies, other types of nomads). Those homeschool out of neccessity, and adjust curriculums and methods of public schools.

I'd say the general consensus in our country is that the goal of a child's education is to prepare it well for its own, independent life. If parents have their own strong ideas of how to do that, they are rightly required to pass some muster, if only to weed out abusers. The parents' rights don't trump the child's.

Non-paywalled link https://archive.is/qe2PJ

Not sure if you consider these anti-Christmas, or just alternative-Christmas, but I like to listen to them around this time.

Bob Neuwirth - Lucky Too https://www.youtube.com/watch?v=gps0W3Wzpwg

Brett Dennen - The Holidays Are Here & We're Still At War https://www.youtube.com/watch?v=1Z7UA5Ykoe0

The Pogues - Fairytale Of New York https://www.youtube.com/watch?v=j9jbdgZidu8

Tim Minchin - White Wine In The Sun https://www.youtube.com/watch?v=fCNvZqpa-7Q

Applying Bernoulli's inequality to the inner term

Inserting into the outer term

You should have used 1807 instead of 7 😉

If we look at the upper blue region:

The area is the sum of the components:

Adding the same (with instead of ) for the lower blue region:

The apothem formula gives

Simple trig says

Those last four equations solve for , or total area of polygon .

As seen in @0xbitcoiner's drawing, the total area is the sum

Therefore,

I think the area of the magenta region is . Let me know if that's correct, and I'll explain.

I guess the question is, are the green lines intersecting chords through one and the same point?

The square roots are all zero because .

So are the integrals, because is an odd function and the intervals are symmetric around 0.

The logs simplify to etc.

Hence, we get

Since , the end result must be

Looks more complicated than it is (or I'm wrong) :-)

Probably surfer slang.