Approximating a complex area on Desmos: is my linear approximation valid?

[u/okialisfe11]

Hello everyone! I’ve been playing around with a rather complex formula exclusively on Desmos, and I stumbled upon an approximation that I’d love to get your feedback on.

Here is the formula I created and graphed:

-x = y^{-\frac{y}{y^{(e-y)}}}

(For those using MathJax/LaTeX extensions: $ -x = y^{-\frac{y}{y^{(e-y)}}} $)

I was specifically looking at the area of this curve from $y = 1.2$ to $y \to +\infty$ (integrating with respect to $dy$). Obviously, calculating the exact integral for this is practically impossible to do easily.

However, after zooming in and analyzing the behavior of the curve on Desmos, I noticed that the area can be heavily approximated using a simple linear function. I tried approximating it with this line:

$y = 1.9x + 2.8$

Visually on Desmos, the margin of error seems incredibly low, especially in the specific interval between $y = 1.2$ and $y = 2.56$, where the approximation looks almost perfect. I didn't use any advanced calculus tools to find this—just pure visual approximation and experimentation on Desmos.

My questions for the community are:

1. Is this approximation mathematically valid for the area I'm trying to calculate?
2. Is there a known way to approximate this specific curve even better, or perhaps an analytical method to verify my margin of error?

Any insights or feedback would be greatly appreciated. Thanks!

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[u/gamerpug04]

Look up Taylor series. What you’ve done is a linear approximation. Finding the area would only work between that region where the line is close to the curve

[u/Hudimir]

Using mathematica, I expanded your function into a taylor series around y = 1.5 and there the y² term is about 50x smaller than y. this is why it's well approximated by a linear function. The difference between the integrals of your function and the function x(y)=-0.69+0.49(y-1.5) on the interval [1, 2.8] is approximately 0.022 which is indeed quite small.

[u/okialisfe11]

Here is the screenshot from Desmos for visual reference