I have frequently used the formula $A=\int_{\theta_1}^{\theta_2} \frac {r^2} {2} d\theta$ when dealing with polar curves, however I never really understood where this formula came from. I'd very much appreciate it if someone would explain it to me in depth.
Thank you all for your time.
Let me try to give you some intuition to the formula.
First consider the case where $r$ is constant, $\theta_1 = 0$ and $\theta_2 = 2\pi$. Then the polar curve is a circle with radius $r$, and you can verify that indeed $A = \pi r^2$.
Now for each $n > 0$, divide the circle into $n$ equal sectors. The area of a circle equals the sum of the areas of these sectors. That is, $$A = \sum_{k=1}^n \frac{\pi r^2}{ n} = \pi r^2.$$ Rewriting this in terms of $\theta = 2 \pi/n$, we get that $$A =\sum_{k=1}^{\frac{2\pi}{\theta}} \theta \frac{r^2}{2}.$$ Now by taking the limit $n \to \infty$ $(\theta \to 0)$, we get $$\int_0^{2\pi}\frac{r^2}{2}d\theta.$$ The story does not change much if $r$ depends on $\theta$, or if $\theta_0$ and $\theta_1$ are different. The idea remains that you are summing an infinite amount of infinitesimally small circle sector areas.