Around the campfire

$$ $C$ is determined by parametric equations $x=\cos t$ , $y=\sin t$ , $-\frac{\pi}{2} \le t \le \frac{\pi}{2}$ , and $dS=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$ , then we obtain $$ $\begin{aligned} N &= \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (2+ \cos^2 t \sin t)\sqrt{(-\sin t)^2+(\cos t)^2}\,dt\\ &=2\left(\int_{0}^{\frac{\pi}{2}}2\,dt-\int_{0}^{\frac{\pi}{2}}\cos^2 t\,d(\cos t)\right)\\ &=2\left(\pi-\left.\frac{1}{3}\cos^3 t\right|_0^{\frac{\pi}{2}}\right)\\ &=2\left(\pi+\frac{1}{3}\right)\\ &\approx \boxed{6.95} \end{aligned}$ $$ $\text{\# }\mathbb{Q}.\mathbb{E}.\mathbb{D}.\text{\#}$