How Do You Calculate Time in the Brachistochrone Problem?

[hriby]

I hope that you've heard about Brachistochrone problem: http://mathworld.wolfram.com/BrachistochroneProblem.html
Given two points, I can find (calculate) the courve, on which the ball needs minimum time to travel from point 1 to point 2.
I get the equation for the courve, which is cycloid, in parametric form, let say:
$$x(\theta)&=&C\left(\theta-\sin{\theta}\right),$$

$$y(\theta)&=&-C\left(1-\cos{\theta}\right).$$

Now I also need to calculate the time needed...
How could I calculate it out of formula below using the equation for the courve/cycloid in parametric form?
$$t_{12}=\int_{T_1}^{T_2} \frac{\sqrt{1+{y'}^2}}{\sqrt{2g\,y}}dx, \quad y'=\frac{dy}{dx}$$
Thanks for your answers!
Hriby

 

[qbert]

how 'bout:

dy/dx = (dy/d_theta)/(dx/d_theta)