Are you bored?

Are you bored?
Do you wanna test your problem solving skills?
well try this problem:
easy mode:
>find the area of the purple region, knowing that the red curve represents y^y=x
hard mode:
>find the area under the curve between a and b, where 1

twis easy, laddy
*dabs*

Do your own homework

i already have the answer, its around 1.3, dont remember the exact answer
also i wish my homewrk would be like this, my actual homework is fucking boring shit

Brainlet here isnt it 1+ sqrt(2) ?

no, how did you get to that solution?

Thats just the first thing that came to mind, I see now its incorrect, the answer is probably sqrt (2)

if you are an engineer, but no

I will admit that I think no method I know will solve this. But I suppose the solution will come in the form "Lambert W something something".

So OP, just share the solution.

its not that complicated, i suggest thinking of it as a geometry problem

Okay. I will ask you just one thing. Does the exact solution come as some combination of usual constants? (Algebraic numbers, pi, e, maybe sine stuff, logarithm stuff, etc.)?

no, in fact i think you cant get the exact solution, but you can calcuate it with a lot of precision.
if it helps think as the curve c:y^y=x as being equal to the curve z:y=l(x), so that l(x) is the inverse of x^x

Oh, fuck you. I took numerical analysis. I can calculate anything with as much precision as I want. But fuck you.

I remember integrating the gauss curve back in high school as a rotational shape and then transforming it back to 2D. is t something along those lines?

i said i think, i dont know if there's an exact answer, but they might be
im not asking for an exact answer tho, its ok if you give an approximation as long as its precise

maybe, my method uses only geometry

Well, you could easily argue that y^y is smaller than y^1.6 and greater than y^1...

between x=1 and x=2

>im not asking for an exact answer tho, its ok if you give an approximation as long as its precise
oh okay then i just happen to know that the curve of that function is the same as as log(x)/lambertw(log(x)) over that region which means the integral to 100 digits of accuracy is 1.328082323968614349068208812545233208407676783192652705757878981412613609057429456035041240934690051

So what is the answer surely someone must know the answer?

Not him but if you are just looking to approximate then that curve looks, locally, very polynom-ish. Just use the 3/8 rule.

You can literally use interpolation to find an arbitrarily accurate solution. "Lmao I never specified it needs to be exact" renders literally any calculation a walk in the park

Quick question. Can the solution be written in terms of the lambert W function?

probably

Write it as a function ffs

can you flip off?

How is this not trivial once you integrate exp(y * log(y)) over some intervals?

Is it possible to integrate y(x) from f(a) to f(b) with great precision using Taylor series and then use basic geometry to find the exact needed area?

thats my method

Brainlet here, why cant you y = sqrt(x) and then integrate that ?

It's not y^2 = x, it's y^y = x

>brainlet here

Oh right.
Well then i guess you will need to use series or numerical integration.

32 and some odd decimal. op should so own math.

This is also boring shit lmao. Mathematicians are autistic I swear.