Hi! I am currently working on some calc2 online homework problems and I am having difficulty with this particular question. To be completely honest I am not sure how to even approach this problem, so if someone would be kind enough to help me solve this one I would really appreciate it, Thank you!
The lemniscate will pass through the origin when $r = 0$, i.e., when $\cos(2\theta) = 0$.This will happen at infinitely many values of $\theta$, but to we want to find two values of $\theta$ between which one loop is traced. Thus we can choose the values of $\theta$: $\theta = \frac{-\pi}{4}$ and $\theta = \frac{\pi}{4}$.
To get the area we then integrate the following:
$A = \displaystyle\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \frac{25}{2}\cos(2\theta) d\theta$
$A = 12.5$