# What is the arclength of (sint/(t+cos2t),cost/(2t)) on t in [pi/12,pi/3]?

Oct 14, 2016

Use $L = {\int}_{\frac{\pi}{12}}^{\frac{\pi}{3}} \sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}} \mathrm{dt} \approx 2.34$

#### Explanation:

Given:
$x = \sin \frac{t}{t + \cos \left(2 t\right)}$
$y = \cos \frac{t}{2} t$

$\frac{\mathrm{dx}}{\mathrm{dt}} = \frac{\sin \left(t\right) \left(2 \sin \left(2 t\right) - 1\right) + \cos \left(t\right) \left(t + \cos \left(2 t\right)\right)}{t + \cos \left(2 t\right)} ^ 2$

$\frac{\mathrm{dy}}{\mathrm{dt}} = - \frac{t \sin \left(t\right) + \cos \left(t\right)}{2 {t}^{2}}$

Arc length with parametric equations

$L = {\int}_{\alpha}^{\beta} \sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}} \mathrm{dt}$

$L = {\int}_{\frac{\pi}{12}}^{\frac{\pi}{3}} \sqrt{{\left(\frac{\sin \left(t\right) \left(2 \sin \left(2 t\right) - 1\right) + \cos \left(t\right) \left(t + \cos \left(2 t\right)\right)}{t + \cos \left(2 t\right)} ^ 2\right)}^{2} + {\left(- \frac{t \sin \left(t\right) + \cos \left(t\right)}{2 {t}^{2}}\right)}^{2}} \mathrm{dt}$

$L \approx 2.34$