Originally Posted by

**topsquark** $\displaystyle L = \int_0^{\pi /2} dt \, \sqrt{ \left ( \frac{dx}{dt} \right ) ^2 + \left ( \frac{dy}{dt} \right )^2 } $

So

$\displaystyle \frac{dx}{dt} = 6sin^2(3t)cos(3t)$

$\displaystyle \frac{dy}{dt} = -6cos^2(3t)sin(3t)$

Thus

$\displaystyle L = \int_0^{\pi /2} dt \, \sqrt{ \left ( 6sin^2(3t)cos(3t) \right ) ^2 + \left ( -6cos^2(3t)sin(3t) \right ) ^2 } $

$\displaystyle L = \int_0^{\pi /2} dt \, \sqrt{ 36sin^4(3t)cos^2(3t) + 36cos^4(3t)sin^2(3t) }$

$\displaystyle L = 6 \int_0^{\pi /2} dt \, \sqrt{ sin^2(3t)cos^2(3t) (sin^2(3t) + cos^2(3t) ) } $

$\displaystyle L = 6 \int_0^{\pi /2} dt \, sin(3t)cos(3t) $

Can you take it from here?

-Dan