I'm stuck because the answer I keep arriving at is negative under a square root sign. In the original question the variable is theta but for ease of typing I'm going to make it x.

If f(x) = 8cos^3(x) and g(x) = 8sin^3(x) , Find the total length of the astroid described by f(x) and g(x) (f(x), g(x)) as x ranges from x=0 to x=2pi

So I found that f'(x) = -24x^2(sin(x^3))

and g'(x) = 24x^2(cos(x^3))

I multiplied through to get:

so int_0^2pi sqrt[ 576x^4(sin(x^6)) + 576x^4(cos(x^6))]

simplified:

24 int_0^2pi x^2 sqrt[sin(x^6) + cos(x^6)]

integrated:

24 int_0^2pi 1/3x^3 sqrt[-cos(x^6) + sin(x^6)]

when I plug in 2pi for x, I get -.0436499839 under the square root sign.

where did I go wrong?

Thanks!