Compute the length of the cardioid below the x-axis?

[IntegrateMe]

$$r = 2sin(\theta)-2$$

First we find x(θ), y(θ)

$$x(\theta) = rcos(\theta)$$
$$y(\theta) = rsin(\theta)$$

Then we find x'(θ) and y'(θ) to use the formula:

$$L = \int_\alpha^β \sqrt{x'(\theta)^2 + y'(\theta)^2} d\theta$$

My problem is that I don't know how to get the limits of integration. The answer key says that they are from 0 to π, but I would have guessed π to 2π, since that represents everything below the x-axis? Any help would be appreciated.

Thanks, guys!

 

[LCKurtz]

Why guess? It's going to cross the x-axis when y = 0. What values of $\theta$ do that? Plot the graph for additional information on what direction it is going when it crosses the x axis.