Question Number 204666 by Ghisom last updated on 25/Feb/24
$$\mathrm{this}\:\mathrm{is}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{curve}: \\ $$$${f}\left(\theta\right)=\left(\mathrm{e}^{\mathrm{i}\theta} \right)^{\left(\mathrm{e}^{\mathrm{i}\theta} \right)} =\mathrm{e}^{−\theta\mathrm{sin}\:\theta} \mathrm{e}^{\mathrm{i}\theta\mathrm{cos}\:\theta} ;\:−\pi<\theta\leqslant\pi \\ $$$${f}:\:\begin{cases}{{x}\left(\theta\right)=\mathrm{e}^{−\theta\mathrm{sin}\:\theta} \mathrm{cos}\:\left(\theta\mathrm{cos}\:\theta\right)}\\{{y}\left(\theta\right)=\mathrm{e}^{−\theta\mathrm{sin}\:\theta} \mathrm{sin}\:\left(\theta\mathrm{cos}\:\theta\right)}\end{cases} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{area} \\ $$