Question

The location of a dot P at a given time t in the x-y plane is given by (x,y) = (t - sin t, 1 - cos t). What is the distance traveled by P in the interval 0 <= t <= 2pi

Answer 1

at time 0 the position is (0,0)

at any time t, the distance wold be,
=sqrt((t-sint)^2+(1-cost)^2)
=sqrt((t-sint)^2+(1-cost)^2)
=sqrt(t^2+2-2(cost+tsint))

now integrate this from 0 to 2pi to get the answer.
wolframalpha

refer to the site to get ur integration done.