Question

Find the area enclosed by the x-axis and the curve x = 4 + et, y = t − t2.

Answer 1

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

$\displaystyle\int_0^1 y(t) x'(t)\,\mathrm dt=\int_0^1 (t-t^2)e^t\,\mathrm dt=\boxed{3-e}$

(you can compute the integral by parts)