Calculus problem #1877

$$ We have $\begin{aligned} \frac{dx}{d\theta}&=3\cos \theta-4\sin\theta\\ \left(\frac{dx}{d\theta}\right)^2&=9 \cos^2\theta+16 \sin^2\theta-12 \sin2\theta \end{aligned}$ and $\begin{aligned} \frac{dy}{d\theta}&=4\cos\theta+3\sin \theta\\ \left(\frac{dy}{d\theta}\right)^2&=16 \cos^2\theta+9 \sin^2\theta+12 \sin2\theta \end{aligned}$ $$ Thus, we obtain $\begin{aligned} L &= \int_{0}^{2\pi} \sqrt{\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2}\,d\theta\\ &= \int_{0}^{2\pi} \sqrt{16(\sin^2\theta+\cos^2\theta)+9(\sin^2\theta+\cos^2\theta)}\,d\theta\\ &=\int_{0}^{2\pi} 5\,d\theta\\ &=10\pi\\ &\approx \boxed{31.416} \end{aligned}$ $$ $\text{\# }\mathbb{Q}.\mathbb{E}.\mathbb{D}.\text{\#}$