매개변수 곡선 길이

$2 \sin 2 t \cos 2 t = \sin 4 t$이므로 $\displaystyle \frac{dx}{dt} = 2 - 2 \cos 4 t$, $\displaystyle \frac{dy}{dt} = 2 \sin 4 t$ 또한 $1 - \cos 4 t = 2 \sin ^{2} 2 t$이므로 구하는 곡선의 길이는 $\displaystyle \int _{0} ^{\frac{3}{8} \pi} \sqrt{\left( \frac{dx}{dt} \right) ^{2} + \left( \frac{dy}{dt} \right) ^{2}} dt$ $\displaystyle = \int _{0} ^{\frac{3}{8} \pi} \sqrt{\left( 2 - 2 \cos 4 t \right) ^{2} + \left( 2 \sin 4 t \right) ^{2}} dt$ $\displaystyle = \int _{0} ^{\frac{3}{8} \pi} \sqrt{8 - 8 \cos 4 t} dt$ $\displaystyle = \int _{0} ^{\frac{3}{8} \pi} \sqrt{8 - 8 \left( 1 - 2 \sin ^{2} 2 t \right)} dt$ $\displaystyle = \int _{0} ^{\frac{3}{8} \pi} \sqrt{16 \sin ^{2} 2 t} dt$ $\displaystyle = \int _{0} ^{\frac{3}{8} \pi} 4 \sin 2 t dt$ ($\because$ $\sin 2 t \geq 0$) $\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} - 2 \cos 2 t \end{array} \right] _{0} ^{\frac{3}{8} \pi}$ $\displaystyle = - 2 \cos \frac{3}{4} \pi - \left( - 2 \cos 0 \right)$ $\displaystyle = 2 + \sqrt{2}$