A Function So Cool That It Wears Shades

Calculus Level 5

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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