Find where the curve intersects the x-axis; this happens when y = 0:
y = t - t ² = 0 → t (1 - t ) = 0 → t = 0, t = 1
Then the area of the bounded region is
[tex]\displaystyle\int_0^1 y(t) x'(t)\,\mathrm dt=\int_0^1 (t-t^2)e^t\,\mathrm dt=\boxed{3-e}[/tex]
(you can compute the integral by parts)
