So I was studying for a test and came across the following prompt:
"What is the total area between the polar curves $r=5\sin(3\theta)$ and $r=8\sin(3\theta)$?"
I guess I'm a bit confused about what the bounds of my integral might look like here. I understand that the setup for the area integral is as follows:
$$\frac{1}{2}\int_a^b(f(\theta)^2-g(\theta)^2)d\theta$$
but I'm a bit confused about how I'd go about finding my bounds for this integral. Any help would be appreciated!
Best Answer
Note that each curve represents a set of three equally spaced pedals with each pedal subtending a polar angle of $\frac\pi3$. Thus, the area is $$\frac{3}{2}\int_0^{\frac\pi3} (f(\theta)^2-g(\theta)^2)d\theta = \frac{3}{2}\int_0^{\frac\pi3} (64\sin^2(3\theta)-25\sin^2(3\theta))d\theta=\frac{39\pi}4 $$