매개변수 곡선 길이

$\displaystyle \frac{dx}{d \theta}$$= - 2 \sin \theta - 2 \sin 2 \theta$, $\displaystyle \frac{dy}{d \theta}$$= 2 \cos \theta + 2 \cos 2 \theta$이므로 $0 \leq \theta \leq \pi$일 때, 이 곡선의 길이는 $\displaystyle \int _{0} ^{\pi} \sqrt{\left( \frac{dx}{d \theta} \right) ^{2} + \left( \frac{dy}{d \theta} \right) ^{2}} d \theta$ $\displaystyle = \int _{0} ^{\pi} \sqrt{\left( - 2 \sin \theta - 2 \sin 2 \theta \right) ^{2} + \left( 2 \cos \theta + 2 \cos 2 \theta \right) ^{2}} d \theta$ $\displaystyle = 2 \int _{0} ^{\pi} \sqrt{2 + 2 \left( \cos \theta \cos 2 \theta + \sin \theta \sin 2 \theta \right)} d \theta$ $\displaystyle = 2 \int _{0} ^{\pi} \sqrt{2 + 2 \cos \theta} d \theta$ $\displaystyle = 2 \int _{0} ^{\pi} \sqrt{4 \cos ^{2} \frac{\theta}{2}} d \theta$ $\displaystyle \left( \begin{array}{l} \begin{matrix} \\ \end{matrix} \end{array} \right.$∵ $\displaystyle \cos ^{2} \frac{\theta}{2} = \frac{1 + \cos \theta}{2}$$\displaystyle \begin{matrix} \\ \end{matrix} )$ $\displaystyle = 4 \int _{0} ^{\pi} \left| \cos \frac{\theta}{2} \right| d \theta$ $\displaystyle = 4 \int _{0} ^{\pi} \cos \frac{\theta}{2} d \theta$ (∵ $0 \leq \theta \leq \pi$) $\displaystyle = 8 \left[ \sin \frac{\theta}{2} \right] _{0} ^{\pi}$$= 8$