# A Function So Cool That It Wears Shades

Calculus Level 4

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

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

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

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

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