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