Optimization and integration

Calculus Level 3

Let \(K\) be the rectangle inscribed in the region bounded by the curve \(y = \cos x\) and the lines \(x = - \pi/2, x =\pi/2\) and \(y=0\) with the largest possible area. Let \(R\) be the region below the curve defined by \(y = \cos x\) but above the \(x\) axis. If \(A\) denotes the area of the region not in the rectangle \(K\) but in the region \(R\), find \(\lfloor 1000A \rfloor\).

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