4. A circle has two parallel chords of length $x$ that are $x$ units apart. If the part of the circle included between the chords has area $2+\pi$, find $x$.
Solution: Let $C$ be the area of the circle, $S$ be the area of the square two of whose edges are the chords, and $A$ be the area of the part of the circle included between the chords. The radius of the circle is $\frac{\sqrt{2}}{2} x$, so $C=\frac{\pi}{2} x^{2}$, and $S=x^{2}$. Then the area $A$ is the area of the square plus one half of the difference between the areas of the circle and square: $A=\frac{C-S}{2}+S=\frac{C+S}{2}=\frac{1+\frac{\pi}{2}}{2} x^{2}$, so $x=\sqrt{\frac{2 A}{1+\frac{\pi}{2}}}=2$.