I am given $x(t) = t^3-17^{2/5}t , y(t) = 2t^2$ $ t \in R$
I found that the curve intersects itself where x=0, which gave me the point $(x,y) = (0, 2*17^{2/5})$ which seems correct.
However I am asked to find the area of the enclosed loop that the parametric curve forms. I am unsure about how to go about this. I read some online about if we have x = f(t), y = g(t), one may find the area using $$\int_{a}^{b} g(t)*f'(t) dx$$ as t increases from a to b. Is this the case for my intersecting loop though? Also when integrating, what would be my limits here? I don't see how I can find for what x the loop has its biggest 'radius'. Hoping for some tips, thanks in advance
There is a difficulty with just going online and looking for an equation for the area. You chose the wrong answer. The equation you found is is fine for an ordinary curve, but yours is not ordinary. It's a closed curve, and that equation is not valid here. In fact, the the equation for the area of a closed curve is given by
$$A=\frac{}{}\int\left(x\frac{dy}{dt}-y\frac{dx}{dt} \right)~dt$$
Now we can tackle the problem at hand. The equations you give are a variation of Tschirenhausen's cubic. For simplicity, let's just write $c=17^{2/5}$. Then,
$$ x(t)=t^3-ct,~ ~ y(t)=2t^2, \quad t\in[-\sqrt{c},\sqrt{c}] $$
Now I'm going to simplify this a bit further by shifting the $y$-axis up so the the curve intersects itself at $x=y=0$. This will not impact the area at all. Then we can write
$$ x(t)=t(t^2-c)\\ y(t)=2(t^2-c) $$
For convenience we can say $t^2-c=T$ for the time being. Now we can proceed to get the area. Thus,
$$ \begin{align} &x(t)=tT\\ &y(t)=2T\\ &\dot x=t\dot T+T\\ &\dot y=2\dot T\\ &x\dot y-y\dot x=tT\cdot 2\dot T-2T(t\dot T+T)=-2T^2=-2(t^2-c)^2\\ &A~=-\frac{1}{2}\int_{-\sqrt{c}}^{\sqrt{c}}2(t^2-c)^2~dt\\ &\quad =-\left[\frac{t^5}{5}-\frac{2ct^2}{3}+c^2t \right]\biggr|_{-\sqrt{c}}^{\sqrt{c}}\\ &\quad =-\frac{8}{15}c^{5/2}\\ &\quad =-\frac{16}{15}17\approx -18.1333\quad \text{for }c=17^{2/5} \end{align} $$
Do not be dismayed by the negative sign. It's a consequence of the curve going in the clockwise direction. I have verified this result numerically by an independent method (integration in the complex plane).