Expert: Sherry Wallin - 12/13/2008

Question
x=(cos(t))^3 y=(sin(t))^3  what is the length of the parametric equation from 0 to
pi? I am more interested in the work than the actual solution thanks.

Answer
Ok C. J.
    Here you are:
If x = cos^3t then dx = 3cos^2t(-sin t) dt or dx/dt = 3cos^2t(-sin t)
so (dx/dt)^2 = 9cos^4t sin^2t
and if y = sin^3t then dy = 3sin^2t cost dt or dy/dt = 3sin^2t cost
and (dy/dt)^2 = 9sin^4tcos^2t
so (dx/dt)^2 + (dy/dt)^2 = 9cos^4t sin^2t + 9sin^4tcos^2
= 9cos^tsin^2t(cos^2t+sin^2t) = 9cos^tsin^2t
sqrt(9cos^tsin^2t)= 3costsint
Now you want to integrate from 0 to pi but you need to make sure that the integral is positive so break it up into two pieces and integrate from [0 to pi/2] (3costsint) dt - integral [pi/2 to pi] (3costsint) dt and then evaluate as you do any integral. I ended up with 3 for an answer.

Math Prof