$$r = 2\cos \theta -1$$
I am suppose to find the polar curve of the inner loop. Here is its graph, courtesy of Wolfram|Alpha,
I am having trouble working out this polar function on a cartesian graph system so my confusion comes from finding the limits of integration for the inner loop. I think if $r = 2\cos \theta -1$ that means I have $r(\theta)$ so $\theta$ is my x and r is my y. So I know that at $\theta = 0$ I have two y values, 0 and 1. How does this work though? Clearly $2\cos (0) -1$ can never be 0. So what is going on here? How does it get two values.
It seems like no one understands my question, I have two values at theta 0. How do I get an arc length if I have two arcs?