How can I calculate an average curve

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I have 5 curves on the same xOy, how can I calculate an average curve for all 5 ?
 
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calculating average points from 5 points does not seem to give correct result.
 
I don't understand what you mean. Do you have 5 points or 5 curves. And what do you mean by "an average curve"- average in what sense?
 
I am sorry it is 5 curves. In my second post, I thought I could pick many points on all of those curves then calculate the average but it didn't turn out correct as expected. Do you know how to draw an average curve ?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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