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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.

# Area Between Curves: Level 3 Challenges

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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