How to solve this cycloid parametric question?

Question

A cycloid is defined by parametric equation x=r(t-sin(t)) and y=r(1-cos(t)) where r is radius of a circle rolling along a flat surface, and the point (x,y) describes where the lowest point on the circle has moved as the circle rolls away from the origin.

1)find dx/dt
2)find dy/dt
3)find dx/dy
4)find length of arc formed by the cycloid from t=0 to t=2pi

Thanks, step by step explaination appreciated.

Answer 1

The answer to 4:
The arclength formula for parmetric equations is
Integral[Sqrt(dx/dt)^2+(dy/dy)^2)]

(dx/dt)^2+(dy/dt)^2 = r^2(1- cos t)^2 + r^2(sin t)^2
[see previous answer for dx/dt and dy/dt]

=r^2(1-2cos t +(cos t)^2 + (sin t)^2)
=r^2(2 - 2cos t)

So the integral becomes
Sqrt(2)r Integral(Sqrt(1- cos t))

I'm not sure how to do the integral. Mathematica gives a value of 4*Sqrt(2) when evaluated from 0 to 2Pi.
So the arclength is Sqrt(2)*r*4*Sqrt(2) = 8r

Answer 2

dx/dt=r{1-cost(t)}

dy/dt=r * sin(t)

You differentiate normally, r is just a constant so it works out nicely.

dy/dx=[dy/dt] / [dx/dt] = cosec(x)-cotan(x)

Arclength is given by the formula:

Integral with limits { Sqrt[1+(f'(x))^2)]}

Which isn't a very nice integral. Give it a bash, you'll probably need some trig identities.