I'm learning how to find the area within polar equations. I know that the integral is: $\frac12\int^{\beta}_{\alpha}f(\theta)^2\ d\theta$. But, how are $\alpha$ and $\beta$ found? Is it something that there is only one method for? Or, are there different strategies to use?
If the answer is that there is just one way, what is it? And, if there is no one-fits-all method, but multiple things to look at - what are they?
Or, is there no good answer for this - and you just need to think about what they could possibly be each time?
Well, what you choose for $\alpha$ and $\beta$ is like asking what you should choose for $a$ and $b$ in a normal Cartesian integral $\int_{a}^{b} f(x)dx$. It depends on what section of $f(\theta)$ you want to integrate, that is, find the area of.
If you want to integrate around the unit circle, integrating from $0$ to $2\pi$ should be suitable. If you want to integrate, for instance, the area contained between two functions, you'd find $\alpha$ and $\beta$ where the functions intersect and integrate that.
Hope this helps