Arc Length, Given Parametric Equations?

Question

Determine the arc length of the asteroid given parametrically by

x(t) = 5 cos^3 (t)
y(t) = 5 sin^3 (t)



PLEASE show me the steps!

answer choices:

15

30

15pi

5pi

5

30pi

I understood the first explanation until...

= 60∫(t=0 to π/2) sin t cos t dt by symmetry

how did it become 60?

The answer below is wrong because sintcost is not sin2t.

sin 2t = 2costsint

helpp!!

Answer 1

Arc length is given by
L = ∫(t = 0 to 2π) √(x'(t)^2 + y'(t)^2) dt
Here x'(t) = -15 cos^2 t sin t and y'(t) = 15 sin^2 t cos t
So x'(t)^2 + y'(t)^2 = 225 cos^4 t sin^2 t + 225 sin^4 t cos^2 t
= 225 sin^2 t cos^2 t (cos^2 t + sin^2 t)
Hence √(x'(t)^2 + y'(t)^2) = 15 |sin t cos t|.
L = ∫(t = 0 to 2π) 15 |sin t cos t| dt
= 60∫(t=0 to π/2) sin t cos t dt by symmetry
= 30∫(t=0 to π/2) sin 2t dt
= 30 [-cos 2t / 2][0 to π/2]
= 30 (0 + 1/2)
= 15.

Answer 2

y=2t-3 dy/dt =2 x=t^2+a million dx/dt=2t dy/dx = dy/dt / dx/dt = 2 / 2t = a million/t (dy/dx)^2 = a million/t^2 a million+(dy/dx)^2 = a million+a million/t^2 sqrt (a million+(dy/dx)^2 ) = sqrt [ a million+ a million/t^2] 0?t?a million t^2=x-a million whilst t=0, x=a million whilst t=a million, x=2 ? sqrt [ a million+ a million / (x-a million) ] dx from a million to 2 ? sqrt [ x / (x-a million) ] dx from a million to 2 This vital seems to diverge whilst x=a million