Chapter 10.3, Problem 51ES

(a) Find the error: The area that is inside the lemnis-cate $r^2=a^2\cos 2\theta$ is

$A={{\displaystyle {\int }_0^{2\pi }\frac{1}{2}{r}^{2}d\theta ={\displaystyle {\int }_0^{2\pi }\frac{1}{2}{a}^2\cos 2\theta d\theta =\left.\frac{1}{4}{a}^2\sin 2\theta \right]}}}_0^{2\pi }=0$

(b) Find the correct area.

(c) Find the area inside the lemniscate $r^2=4\cos 2\theta$ and outside the circle $r^2=\sqrt{2}.$