Question

Find the length of the loop of the given curve.
x = 3t - t^3
y = 3t^22

I know the formula, I just dont remember how to find the limits for t.

Answer 1

http://www.wolframalpha.com/input/?i=%28x%3D3t-t%5E3%29++%28y%3D3t%5E22%29

Answer 2

look in the alternate forms tab

Answer 3

that gives me zeros but not the closed interval. Am I missing something obvious?

Answer 4

if this is a closed look, then we know that at t = a and b, (x,y)(t) is the same. So in particular, give the form of y(t), it must be that a = -b

Now given that, is it the case that
x(a) = x(b)?

I.e., x(a) = x(-a)
=> 3a - a^3 = -3a + a^3
=> 2a^3 - 6a = 0
=> 2a (a^2 - 3) = 0

Hence the loop closes over the interval in t
\[[-\sqrt{3}, \sqrt{3}]\]

Answer 5

(Note that a = 0 is not a solution for us because that gives us no pair of points in the y-ordinate function.)