Find length of the curves

Question

Find the length of the curve x =sin^3 t, y=cos^3 t,

0<=t<= 2 pi. Note thata the integrand of the integral representing the arc length is always larger than or equal to 0

Answer 1

The length formula is

L = ∫ sqrt [ (dx/dt)² + (dy/dt)² ] dt with limits 0 to 2 pi

After the power rule and some algebra this leads to

L = 3 ∫ sqrt [ sin²(t) cos²(t) ] dt with limits 0 to 2pi

Since the integrand is periodic with period pi/2 :

L = 12 ∫ sqrt[ sin²(t) cos²(t) ] with limits 0 to pi/2

The double angle formula can be used to rewrite this as

L = 6 ∫ sqrt[ sin² (2t) } dt with limits 0 to pi/2

Since the argument of the square root is now always positive this is

L = 6 ∫ sin(2t) dt with limits 0 to pi/2

Simple calculus can be used to evaluate this resulting in

L = 6