Solution:

Not many were able to solve the length of the cycloid...
Don't worry, i never solved this one myself!!! Had an hour of frustration before looking at the solution...

Here's my version of the solution:
If you are into center of rotation, you can easily write the velocity as a function of time:
Here, the length CP is 2RCos(90-theta), which is 2RSinQ..
Q is infact wt...so V=[2RSin(wt)]*w...as the velocity is equal to the distance from center of rotation*angular velocity..

All that remains is to integrate V(t) from 0 to 2pi/w...