Calculus
# Area Between Curves

*cute little pillowed diamond* in the middle be \(A\). What is the value of \(\lfloor 1000A\rfloor \)?

A square has vertices at \((1,1), (-1,1), (-1,-1), \text{ and } (1,-1).\)

Let \(S\) be the region of all points inside the square which are nearer to the origin than to any edge. What is the area of \(S\)?

\(\triangle ABC\) is constructed such that \(AB=3\), \(BC=4\), and \(\angle ABC=90^{\circ}\). Point \(P\) is chosen inside \(\triangle ABC\), and points \(E\) and \(F\) are drawn such that they form line segments with \(P\) that are perpendicular to sides \(AB\) and \(BC\), respectively. If \(\mathbf{L}\) is the locus of all points \(P\) such that \([PEBF]\ge 1\), then find the value of \[\left\lfloor 1000\dfrac{[\mathbf{L}]}{[ABC]}\right\rfloor\]

\(\text{Details and Assumptions:}\)

\([\mathbf{N}]\) means the area of the locus \(\mathbf{N}\), and \([PQRS]\) means the area of \(PQRS\). \(\lfloor x \rfloor\) is the floor function.

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