Area Between Curves

Area Between Curves: Level 3 Challenges


The area of triangles enclosed by the axis and the tangent to y=1xy=\dfrac1x is always constant, no matter what xx you pick. Find the value of this area.

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

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

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

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

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

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

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


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