Strict Linear Order on Plane: Definition & Geometric Description

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Homework Statement


Define a relation on the plane by setting (x_0, y_0) < (x_1, y_1) if either

y_0 - x_0^2 &lt; y_1 - x_1^2, or y_0 - x_0^2 = y_1 - x_1^2 and x_0 &lt; x_1

I showed that this is a strict linear order on the plane but then the question asks me to describe it geometrically. I am not really sure how you describe something like this geometrically. Anyone want to clarify what that means?

Homework Equations


The Attempt at a Solution

 
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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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