A Function So Cool That It Wears Shades

Calculus Level 4

There exists a unique, positive-valued, non-constant, continuous and differentiable function y=f(x)y = f(x) such that

(i) over any specified interval, the area between f(x)f(x) and the xx-axis is equal to the arclength of the curve, and

(ii) f(0)=1f(0) = 1.

If S=12f(x)dxS = \displaystyle\int_{-1}^{2} f(x) dx, then find 1000S\lfloor 1000S \rfloor.

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