Areas shared by polar regions problem

[shivers20]

1. Shared by the circle r=1 and r=2 sin theta

Sketch a graph: I get three circles. r=1 is one circle. I must have messed up on plotting points. Maybe you can tell me how to plot them correctly?

theta| 0 pi/2 pi 3pi/2 2pi
---------------------------------- repeats
r | 0 2 0 -2 0

Solve 2 sin theta = 1; sin theta = 1/2

so, d= pi/6 and Alpha = 2pi-pi/3

Use top/bottom symmetry

A= 2 (pi/6 to pi) 1/2 [(2sin theta)^2 -1^2] d theta

(4 sin^2 theta -1) d theta

I forgot how to integrate the rest.

 

[galactus]

Find the limits of integration by equating functions:

\(\displaystyle 2sin({\theta})=1\Rightarrow{sin^{-1}(\frac{1}{2})={\theta}\)

\(\displaystyle {\theta}=\frac{\pi}{6} \;\ and \;\ \frac{5{\pi}}{6}\)

\(\displaystyle \L\\\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{5{\pi}}{6}}(4sin^{2}({\theta})-1)d{\theta}\)

\(\displaystyle \L\\\int{sin^{2}({\theta}))d{\theta}=\frac{1}{2}{\theta}-\frac{1}{4}sin(2{\theta})+C\)