Question

find the area of the region that lies inside the cardiod r=1+cos(theta) and the outside of the circle r=1

Answer 1

\[S=\int rdrd\theta=\int\limits_{-\pi/2}^{\pi/2}\left(\int\limits_{1}^{1+\cos\theta} rdr\right)d\theta=
\frac{1}{2}\int\limits_{-\pi/2}^{\pi/2}\left((1+\cos\theta)^2-1\right)d\theta=\]
\[=\frac{1}{2}\int\limits_{-\pi/2}^{\pi/2}\left(2\cos\theta+\cos^2\theta\right)d\theta=\int\limits_{-\pi/2}^{\pi/2}\cos\theta d\theta+\frac{1}{2}\int\limits_{-\pi/2}^{\pi/2}\cos^2\theta d\theta=\]
\[=2+\frac{1}{2}\cdot\frac{\pi}{2}=2+\frac{\pi}{4}\]