The curve hits the y-axis whenever \(x=0\):
\(x=t^2-2t=0\implies t(t-2)=0\implies t=0,t=2\)
The area enclosed by the region is the sum of distances of every point on the curve where \(t\in[0,2]\) to the origin. This is given by the integral
\(\displaystyle\int_0^2\sqrt{x(t)^2+y(t)^2}\,\mathrm dt\)
\(\displaystyle\int_0^2\sqrt{(t^2-2t)^2+t^2}\,\mathrm dt\)
\(\displaystyle\int_0^2\sqrt{t^2(t-2)^2}\,\mathrm dt\)
\(\displaystyle\int_0^2|t||t-2|\,\mathrm dt\)
Since \(t\in[0,2]\), you have \(|t|=t\) and \(|t-2|=-(t-2)=2-t\), giving you
\(\displaystyle\int_0^2(2t-t^2)\,\mathrm dt=t^2-\dfrac{t^3}3\bigg|_{t=0}^{t=2}=4-\dfrac83=\dfrac43\)
