# Inscribed circles under partial circles

**Calculus**Level 5

We have a semi circle of radius 1 with diameter \(AB\). A line \(AC\) is drawn making an angle \(\theta\) with \(AB\) and intersecting the semicircle at \(C\).

Now a circle is drawn such that it touches line \(AB, AC\) and the arc \(BC\) inside the semicircle.

Let the radius of the circle be \(r\), if \(r\) can be written as \( r= f(\theta) \), find \( \displaystyle 6\int _{ 0 }^{ 1/2 }{ f(2\tan ^{ -1 }{ \theta )d\theta } } \).