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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 5 Challenges

The ends of a stiff bar $$\overline{AB}$$ of length 4 slide freely inside a parabolic track (specifically, the parabola $$y=x^2$$). As they do, the midpoint $$M$$ of that bar traces a curve. Find the area of the region between the parabola and the curve traced by $$M$$.

Assume that the bar slides infinitely in both directions.

If the average distance from a point randomly selected in the unit square to its center equals $\frac{\ln(a+\sqrt{b})+\sqrt{b}}{c}$ where $$a,b$$ and $$c$$ are positive integers with $$b$$ being square-free. Find $$a+b+c$$.

Let $$P$$ be a point (other than the origin) lying on the parabola $$y = x^{2}$$. The normal line to the parabola at $$P$$ will intersect the parabola at another point $$Q$$. The minimum possible value for the area bounded by the line $$PQ$$ and the parabola is $$\dfrac{a}{b}$$, where $$a$$ and $$b$$ are positive coprime integers. Find $$a + b$$.

Clarification: The normal line is the line perpendicular to the tangent line at a given point on a curve and which passes through the given point.

I have a infinitely long strip of paper that is one unit wide. On the paper is a point $$P$$ that is $$\dfrac{1}{4}$$ units away from one of the edges of the strip.

Let $$L$$ be the locus of all points $$O$$ such that if I draw the circle with center $$O$$ passing through $$P$$, the entire circle can be drawn on the strip of paper. If the area of $$L$$ can be expressed by $$\dfrac{a\sqrt{b}}{c}$$ for relatively prime $$a,c$$ and square-free $$b$$, then find $$a+b+c$$.

A square $$5 \times 5$$ is positioned in the $$4^\text{th}$$ quadrant intersecting part of a circle of radius $$5$$ and centered at $$(-2, 0)$$. The square is rotated counter-clockwise about the origin.

Through what angle (in degrees) does the square have to be rotated until $$\frac 1 2$$ of its area intersects with the circle?

[Answer to nearest 3 decimal places]

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