Find the area enclosed by the curve y = x csch(x+y)

[Saracen Rue]

TL;DR Summary

Determine the area enclosed by the curve $y=x \text{ csch}(x+y)$ and the positive $x$-axis

The title and summary pretty much say it all. I was wondering if it's possible to accurately determine the area enclosed by the curve $y=x \text{ csch}(x+y)$ and the $x$-axis?

I first tried solving for $y$ and then $x$, however it doesn't appear possible to solve for either variable. I then recognised that the equation can be represented as a nested function, such that: $$f(x)=x \text{ csch}(x+x\text{ csch}(x+x\text{ csch}(x+x\text{ csch}(x+...))))$$

Through this I was able to use Desmos Graphing calculator to help me estimate the enclosed area to by integrating $f(x)$ from $0$ to $\infty$. However, the imbeded nature of the function made it hard to properly find a definitive solution and I could only gather that the area appears to be approaching $1.5$ square units, although this might be inaccurate and it could be simply approaching a number near $1.5$.

How would I go about accurately determining this area?
Any help is greatly appreciated.

 

[Mark44]

Summary:: Determine the area enclosed by the curve $y=x \text{ csch}(x+y)$ and the positive $x$-axis

The title and summary pretty much say it all. I was wondering if it's possible to accurately determine the area enclosed by the curve $y=x \text{ csch}(x+y)$ and the $x$-axis?

I first tried solving for $y$ and then $x$, however it doesn't appear possible to solve for either variable. I then recognised that the equation can be represented as a nested function, such that: $$f(x)=x \text{ csch}(x+x\text{ csch}(x+x\text{ csch}(x+x\text{ csch}(x+...))))$$

Through this I was able to use Desmos Graphing calculator to help me estimate the enclosed area to by integrating $f(x)$ from $0$ to $\infty$. However, the imbeded nature of the function made it hard to properly find a definitive solution and I could only gather that the area appears to be approaching $1.5$ square units, although this might be inaccurate and it could be simply approaching a number near $1.5$.

How would I go about accurately determining this area?
Any help is greatly appreciated.

Your equation is equivalent to $F(x, y) = 0$, with $F(x, y) = x - xcsch(x + y)$. As such, the graph of F is a surface in $\mathbb R^3$, and the equation you're working with is the cross-section of the surface in the x-y plane, i.e., where z = 0.
As far as how you would graph F(x, y) = 0, nothing comes to mind, except possibly calculating function values at a bunch of points. There are going to be discontinuities at points where $x + y = k\pi$, k an integer.

 

[BvU]

As far as how you would graph F(x, y) = 0, nothing comes to mind

I think that was already done. Wolfram plots:

Re: points where x+y=kπ ? it's hyperbolic: $$y = {x\over \sinh(x+y)}$$

 

[Saracen Rue]

I think that was already done. Wolfram plots:
View attachment 261445Re: points where x+y=kπ ? it's hyperbolic: $$y = {x\over \sinh(x+y)}$$

Sorry for not being clear in the OP. You're right, I do already have the graph of $y=x \text{ csch}(x+y)$ and I'm just trying to find the area enclosed by the cruve and the $x$-axis

Edit: Do you think it would be possible to use WolframAlpha's nest function to create and integrate a heavily nested form of $y=x\text{ csch}(x+x\text{ csch}(x+...))$ to allow for a better estimation of the area? I tried doing it myself but I'm not the most familiar with the language and could only seem to get it to nest every variable $x$ with $y(x)$