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Area Between Curves

In the language of Calculus, the area between two curves is essentially a difference of integrals. The applications of this calculation range from macroeconomic tax models to signal analysis.

Level 3

         

The area of triangles enclosed by the axis and the tangent to \(y=\dfrac1x\) is always constant, no matter what \(x\) you pick. Find the value of this area.

Above is the graph of the equation \[|x|+|y|-|x||y|=0.5\] Let the area of the cute little pillowed diamond in the middle be \(A\). What is the value of \(\lfloor 1000A\rfloor \)?

Find the area bounded by the curve \(f(x)=x+\sin(x)\) and its inverse function between the values \(x=0\) and \(x=2\pi\).

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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