How to find limits of integration when calculating length of loop?

Question

for the parameterization:

x = 6t – 2t^3

y = 6t^2

calculate the length of this curve's loop.

I know the formula for doing this, but since (a ≤ t ≤ b) is not given here nor is a graphical representation of the curve (graphing calculator not allowed), I don't know what the limits of integration should be.

Knowing where on the xy-plane the loop starts and ends is probably helpful, but I don't know how to determine where this happens.

Answer 1

You need to find $t_1, t_2$ such that $x(t_1) = x(t_2)$ and $y(t_1) = y(t_2)$

$y(t_1) = y(t_2) \implies t_1 = -t_2$

$x(t)$ is odd (with the $t_1 = -t_2$ above) $\implies x(t_1) = 0$

$6t_1 - 2t_1^3 = 0\\ t_1(t_1^2 - 3) = 0\\ t_1 = \sqrt 3\\ t_2 = -\sqrt 3$

$\int_{-\sqrt 3}^{\sqrt3} \sqrt {\frac {dx}{dt}^2 + \frac {dy}{dt}^2} \ dt$