# Leave me please!

**Calculus**Level 5

Mayank creates a magical box of side \(s\). Meanwhile, Akul caught a fly and tied it to an ideal thread. Now, the free end of thread is tied to the center of cube and the setup is left as such.

**Why is the box magical?**

The box is such that the fly can fly freely inside the cube if \(d<\dfrac { s }{ 2 } \), where \(d\) is the distance of the fly from the center.

As soon as its distance from the center exceeds \( \dfrac { s }{ 2 } \), it will find itself outside the cube with the same distance from the center and can again enter the cube only when its distance from center becomes \( \dfrac { s }{ 2 } \).

Also as it just enters the cube at d=\( \dfrac { s }{ 2 } \), It has an equal probability to be at any point at a distance \( \dfrac { s }{ 2 } \) from the center.

Find the expected distance of the fly from the center of the cube given that the length of thread is \(\dfrac { s }{ \sqrt { 2 } } \).

If it is equal to \(k\times s\), find \( \left\lfloor 10^4\times k \right\rfloor \).

**Details and assumptions**:

The fly doesn't feel any tension because of the magical thread.

The fly can fly!