Area under the curve equals to the arc length?

This function is special because its area under the curve is the same as the arc length!

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Thank you,
blackpenredpen

[precisionman]

First reflex when seeing the title : "stupid question, y=1"

[srpenguinbr]

i did it in a different way:
y=sqrt((y')²+1) because of those integrals
y²=y'²+1
i took the derivative in both sides to get rid of the second powers:
2yy'=2y'y''
if y'=0, we get y=c and substitute in the original differential eq to find out that c=1
else, we have y=y'' which has the solutions y=c1e^x+c2e^-x. Substitute in the original eq to get c2=1/(4c1).

It is not exactly equal to cosh (x+c) for all c, but you can get the same functions using other values for the constants

[angelmendez-rivera]

For those complaining that this is impossible because an area cannot be equal to a length, let me just say that they can indeed be. This is mathematics, not physics. Area and length im mathematics have very precise definitions that are just different from the definitions in physics. The area of a set is the Lebesgue measure of set with topological dimension 2, and the length of a set is the Lebesgue measure of a set with topological dimension 1.

[muttleycrew]

Loved this question. I wrote this DE as y^2 - (dy/dx)^2 = 1 which is clearly a hyperbolic form of DE, _immediately_ suggesting one of the hyperbolic trig functions should offer a possible solution and the moment you try y = cosh(x) it works out.

[rateeshk]

Nice effects in the video
bprp with vfx deserves more likes

[ДаниилРабинович-бп]

It was assumed that the function is analytic. Example of many possible additional solutions: cosh(x) for x<0, 1 for 0<=x<=1, cosh(x-1) for x>1

[danielnelson]

I would love to see more videos about cool properties with hyperbolic functions. They always fascinate me

[Victor_Gonzalez]

*me to my calc prof*: 3:04 just keep the positive, it's much better

*my calc prof*: maybe in september

[giorgiomicaglio]

Wait. This is 1080p 60 fps, wonderful😍

[pawanshkl]

Please bring a marathon class on complex analysis

[emir]

0:17 that only moment he removed the Teddy Bear so we can read what's written on his t-shirt

[Taterzz]

MindYourDecisions had something similar from a problem asking about slack on a bridge. it's pretty insane how the number e just manages to show up everywhere.

[_DD_]

This is cool conceptually. 👍 I also like problems where you have to maximize the area of one object inside another one, if you know what I mean. Like the biggest volume of sphere inside a conus and similar.

[kepler]

The moment I saw this video, I immediately thought of coshx

[ligleaper]

Maybe you can explain why bridge builders use hyperbolic trig functions in their geometry?

[mathphschjhb]

not only e^x is Unique, cosh x also! That’s brilliant man!

[natolio]

The integral of e^-i*x(y^2) + e^i - x/(y-400) over the interval of a to a. I believe the area is, in fact, equal to the arc length.

[sarthaksrivastava]

Your videos are great and only because of these I have managed to become good at calculus ;)
also, I made a twitter account just to send you an integral based question (if you remember I tried asking you how to send it ) hope you see into it.

[noahtaul]

Here’s a silly example: f(x)=1 for x<0 and f(x)=cosh(x) for x>=0. This is a differentiable function

[dominicellis]

I equated the two arguments then solved the differential equation f(x) = z, z = sqrt(1+zdot^2). From here square both sides and solve for zdot in terms of z. Then you integrate using a trig substitution. You’ll end up with theta = iacos(z) and using the complex definition of cosine you can easily derive that z = cosh(x) and that this functions indeed satisfies the differential equation.