Note: Assume that any solutions are limited to $$0\leq\theta\leq2\pi.$$ How do you consistently find polar areas without graphing the given curve(s)? I.e, suppose we wish to find the area between the loops of $$r=4+8\cos(\theta).$$ Setting the expression equal to zero in order to find points of intersection, we get $$ 4+8\cos(\theta)=0 8\cos(\theta)=-4 \cos(\theta)=\frac{-1}{2} \theta=\frac{2\pi}{3}, \frac{4\pi}{3} .$$ I am confused about how to algebraically find the necessary integral(s) to solve for the correct area, which is $133.404$.
Consider another problem. Suppose we wish to find the common interior of $r=4(1+\sin(\theta))$ and $r=4(1-\sin(\theta))$. Setting the expressions equal, we get that $\theta=0$ and $\theta=\pi$. I am still at a dilemma though, it being figuring out a reliable method to setup integrals without graphing the equations.
I would appreciate an explanation. From my understanding, the problems can be only so varying--either it's inside one shape and outside another or inside of both shapes.