I have a question about the area enclosed between the following parametric equations:
\begin{align*}
x &= t^3 – 8t \\
y &= 6t^2
\end{align*}
I know the area is the integral of the $y(t)$ times the derivative of $x(t)$. What I don't know is how to find the limits of integration for $t$.
Thank you!
Best Answer
by drawing a graph, e.g.
http://www.wolframalpha.com/input/?i=draw+x+%3D+t%5E3-8t,++y+%3D+6t%5E2
you can see that the loop is around points where $x = 0, y \ne 0$, that is $ t^3 - 8t = 0, t = +/- \sqrt8$, these are your limits, then as you said
$A = \int\limits_{-\sqrt8}^{\sqrt8} y(t) x'(t) dt = 1303.3...$