Given the following two parametric equations:
$$x = t^2$$
$$y = t^3 -t$$
how can I find the points of where those two lines intersect?
I am asked to find the area of the enclosed region and so far I got
the integral figured out now I just need the limits of integration
which should be where these two lines meet, right?
Best Answer
Suppose that $t_1$ and $t_2$ are the parameters of a self-intersection point. In other words, $$ t_1^2 = t_2^2 \qquad \text{and} \qquad t_1^3 - t_1 = t_2^3 - t_2. $$ The first equation (equating the $x$-coordinates) shows that $t_2 = \pm t_1$, so if we want distinct parameters $$ t_2 = -t_1. $$ Then, the second equation (equating the $y$-coordinates) becomes $$ \begin{align} t_1^3 - t_1 &= \left( -t_1 \right)^3 - \left( -t_1 \right) \\ &= - t_1^3 + t_1 \\ &= - \left( t_1^3 - t_1 \right), \end{align} $$ which shows that $$ 0 = 2(t_1^3 - t_1) = 2t_1(t_1 - 1)(t_1 + 1). $$ Hence, $t_1 = \pm 1$ with $t_2 = - t_1 = \mp 1$. Therefore, the self-intersection point is $$ (x, y) = (1, 0). $$