If x = 18cos^3θ and y = 18sin^3θ, find the total length of the curve swept out by the point (x,y) as θ ranges from 0 to 2π.

Answer 1

just plug and chug:

s = ∫[0,2π] √(x'^2 + y'^2) dθ
Using symmetry, we can take just one arc, and
s = 4∫[0,π/2] √((-54cos^2θsinθ)^2 + (54sin^2θcosθ)^2) dθ

this simplifies quite nicely