Optimization and integration

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 $\lfloor 1000A \rfloor$ .