I am working on this problem Evaluating the area between the curves $r=2\sin\theta$ and $r=\sin\theta+\cos\theta$.
I need to find the area of the larger circle outside the smaller circle. It is enough to find area of the part of a smaller circle which is inside the larger circle. My doubt is to how to find the limits for $\theta$. Or, is there any alternative way to do the problem?
Please note that both $r = \sin\theta + \cos\theta$ and $r = 2 \sin\theta ~ $ circles form between $0 \leq \theta \leq \pi$ and to find their intersection point,
$r = \sin\theta + \cos\theta = 2 \sin\theta \implies \theta = \pi /4$
So one of the approaches will be to find the area bound by individual curves as in the below diagram and then subtract the smaller area from the larger area.
Please note that the section of $r \leq \sin\theta + \cos\theta$ (smaller circle) shown in the diagram forms for $\pi/4 \leq \theta \leq 3\pi/4$ whereas the section of $r \leq 2 \sin\theta$ (larger circle) shown in the diagram forms for $\pi/4 \leq \theta \leq \pi$.
So the integral can be written as,
$A_l = \displaystyle \frac{1}{2} \int_{\pi/4}^{\pi} (2 \sin \theta)^2 ~ d\theta$
$A_s = \displaystyle \frac{1}{2} \int_{\pi/4}^{3\pi/4} (\sin \theta + \cos\theta)^2 ~ d\theta$
and desired area $A = A_l - A_s$.